Bowing Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣸⡿⢁⣤⠤⣄⠀⠈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⣠⠤⣤⡈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠸⣇⣀⡼⠀⢀⣀⡀⠙⢿⣆⠀⠀⠀⠀⣰⡿⠋⢀⣀⡀⠀⢧⣀⣸⠇⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠈⠉⠀⢰⡏⠉⢹⠄⠈⣿⣿⠿⠿⢿⣿⠁⠠⡏⠉⢹⡆⠀⠉⠁⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠙⠒⠋⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠙⠒⠋⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⢀⡴⠒⢦⡀⠀⣸⣿⠀⠀⢸⣿⠀⢀⡴⠒⢦⡀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣠⣤⣀⠘⢧⣤⡼⠁⣠⣿⠿⠿⠿⠿⣿⣄⠈⢧⣤⡼⠃⢀⣤⣄⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣹⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⣏⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⡉⠉⠁⣀⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⠈⠙⢉⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠁⠀⠈⠉⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠁⠀⠈⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢤⣄⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⣠⡤⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠈⠙⠶⣤⡀⠀⠙⣿⣶⣶⣶⣶⣿⠋⠀⢀⣤⠶⠋⠁⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠉⠀⠀⢹⣿⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⢀⣠⡤⠖⠛⠀⢀⣿⣿⣶⣶⣾⣿⡀⠀⠛⠲⢤⣄⡀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠚⠉⠁⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠈⠉⠓⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣶⣶⠿⠋

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣛⣿⣛⣟⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣿⠃⠈⠙⠻⠋⠙⢿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡋⠀⠭⢠⠄⠭⠀⠈⣣⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠘⡂⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢾⡳⣦⡄⠡⣤⠖⣿⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠁⠀⠀⠀⠈⠙⠛⠋⠉⢙⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⢀⠀⠘⣧⢈⣽⢾⡅⣰⡇⠀⡀⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⢈⣿⡄⠀⠛⠀⡀⠘⠃⠀⣼⠋⠁
⠀⠀⠀⠀⠀⠀⢠⡏⢸⡆⠀⢈⡷⠀⢿⠋⡿⢾⡏⣻⠀⢰⡃⠀⠀⡾⠈⡇⠀⠀⠀⠀⠀⣾⣿⣿⣦⣀⠀⠒⢀⣠⣾⣁
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠧⠀⢸⡄⠀⠸⣾⠇⠀⣷⡏⠀⠀⡿⠀⠰⠇⠀⣷⠀⠀⠀⢀⡴⠛⠉⡁⠀⡏⠉⠈⠉⣿⠀⣈⠙⠲⢤
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⡀⠀⢳⡀⠀⣿⠀⠀⣻⠁⠀⣸⠃⠀⣴⠀⠀⢿⠀⠀⠀⣸⠈⡇⢀⡽⠀⡇⠀⠀⠀⣿⠰⢯⠀⡿⠸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠈⢷⡀⢸⡆⢀⡟⠀⣰⠃⠀⠀⡟⠀⠀⢸⡀⠀⠀⣿⠀⢩⠈⣇⠀⢻⡶⠶⢶⡇⠀⣼⢀⡁⠀⡇
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⣇⠀⠀⠀⠳⡄⢷⣸⢃⡼⠃⠀⠀⢰⡇⠀⠀⢸⡇⠀⠀⡏⠀⢸⠀⠸⡄⠘⣇⠀⡿⠀⣰⠃⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣦⠀⠀⠀⠀⠙⠈⠁⠈⠀⠀⠀⠀⣠⠆⠀⠀⢸⡇⠀⢀⡇⠀⠸⡆⠀⠹⣄⢹⣼⠇⣰⠋⠀⢸⠃⠀⣿
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⢳⡀⠀⠀⠀⠶⡽⠀⠀⠀⠀⡴⠃⠀⠀⠀⢸⡇⠀⢸⡇⠀⢀⡷⠀⠀⠘⠋⠋⠘⠁⠀⠀⣏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠹⣄⠀⠀⢰⡆⠀⠀⢀⡞⠁⠀⠀⠀⣠⠞⠀⠀⢸⡇⠀⠀⠹⣄⠀⠀⢠⣹⡆⠀⠀⣴⠋⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢘⣧⠀⠸⡇⠀⣰⣏⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠳⣄⠀⠀⠘⢦⠀⠀⣥⠀⢀⡞⠁⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣆⢠⡞⣩⠼⠷⢤⣤⠶⠿⢬⡳⣄⢠⠞⢻⠀⠀⠀⠀⠀⠀⢹⣦⡀⢠⡼⣷⣀⠋⣠⡿⢤⡀⢀⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠾⡇⠀⣴⢋⠀⠀⠀⢀⡷⠟⠋⠀⢸⡀⠀⠀⠀⠀⠀⣼⠈⠳⢯⡎⢁⣽⡋⠁⠈⣷⠷⠋⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠹⣆⡿⣫⢆⠀⢠⠞⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠹⣼⢟⡥⣀⡼⠃⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣕⣵⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡾⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⡡⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡤⣿⣿⠃⠀⠀⠀⣀⠀⠀⢀⣩⠥⠚⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠃⠱⠇⠀⠐⣀⣩⡤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃⢈⣠⣤⢶⡞⠋⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣊⣡⣴⣾⣿⣿⣧⣾⣷⠃⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣿⣿⠟⣹⣿⡏⣻⢿⣿⣿⠃⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣵⡿⠋⠁⢠⣿⡿⢀⣷⣿⣿⠃⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⣸⡿⠋⠀⠀⠀⣾⡽⠃⡌⢻⠿⠁⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⢀⣾⠋⠀⠀⢀⣤⣤⣯⣶⣼⣤⣁⡀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣠⡿⠁⠀⠀⠀⢫⣶⣄⣠⠉⢙⠛⠙⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣰⡟⠀⠀⠀⠀⠆⠋⣉⡛⠻⠷⣿⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢰⡏⢿⡡⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⢸⠁⠦⣶⡌⠇⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠰⡆⡄⣄⣸⣿⡏⠀⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⢀⣴⢆⡇⣿⢀⣿⣿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠇⠀⠀⠀⠀⠀⠀⡌⢰⢻⡟⣷⣾⣿⠇⠀⠀⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡏⠀⠀⠀⠀⠀⠀⠘⠼⣻⣇⡧⢹⣿⡟⠀⠀⠀⡏⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣿⠀⠀⠀⠀⠀⠀⠀⠀⢎⢽⢩⣆⡿⣻⣇⠀⠀⠀⢁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢰⡏⠀⠀⠀⠀⠀⠀⢀⡴⢸⣞⡄⣸⣿⠛⢋⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⠐⠁⠀⠀⠀⠀⡜⡻⢫⢴⣿⡏⠀⠘⢃⠀⠀⠀⢦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡠⠂⠀⠀⠀⠀⡀⣦⣹⢻⣌⣿⣿⠁⠀⠘⣎⢢⡀⠀⢸⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣷⡧⠊⠀⠀⠀⠀⠀⠀⢥⣞⣵⣋⣼⣿⣧⣤⣄⣀⣈⣙⣿⣷⣮⣧⣵⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢐⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⢿⣟⣟⣛⡿⠯⠽⠂
⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠀⠄⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⡀⢀⣀⣄⠀⡀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⣀⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠒⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⠄⠀⠀⠀⠀⣤⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀
⠀⢀⠂⠁⣀⣤⢠⣤⢤⣤⣀⠀⠀⠀⠀⠀⢤⣾⣧⠀⠀⠀⠀⠀⢸⣿⠦⠄⠀⠠⠤⢀⣀⣀⣀⠤⠤⢤⣄⣀⠁⢀
⠀⠈⡗⠉⠉⣀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢤⡀⠉⠙⡆
⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⡀⠀⠀⠀⣨⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠚⠋⠝⠉⠀⠀⠀⠈⠉⠙⢓⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠙⠙⠳⢤⡀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⡈⠀⣠⠈⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠆⠘⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢲⣦⣾⣷⣦⡀⣀⢀⣴⣾⣷⣶⡆⠀⠀⠀⢀⣶⣷⣦⣄⣀⣴⣾⣶⡀⠀⠀⠀⠀⢠⣤⣤⣀⠀⢀⣠⣤⣤⡄⠀⠀⠀⢀⣴⡿⠿⣷⡀⢰⣾⠿⢷⣄
⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠘⣿⣿⡿⡯⢿⢿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⠿⢿⣿⣿⣿⠇⠀⠀⠀⠈⣿⣀⣀⣨⠻⢫⣀⣀⣸⡯
⠀⠀⠀⠚⠛⠻⠟⠋⠀⠀⠀⠙⠻⠟⠛⠓⠀⠀⠀⠀⠈⢡⣿⣧⢸⣿⢿⠁⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠘⢻⣿⡿⠀⠘⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠁⠀⠀⠀⠀⠙⠿⠆
⠀⠀⠀⠀⢠⣶⣿⣷⡀⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⢀⣤⣦⠀⠀⠀⠀⠀⠀⢰⣶⣦⡀⠀⠀⣠⣴⣶⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠸⣿⣄⡈⢳⡛⠁⣨⣿⡇⠀⠀⠀⠀⠀⠀⣹⣿⣿⣶⣾⣿⣿⡆⠀⠀⠀⠀⠀⠻⢿⣿⣿⣦⣼⣿⣿⡿⠇⠀⠀⠀⢰⣿⣿⣿⣶⣤⣤⣴⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠈⣩⣿⡟⢻⣿⡋⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢿⣷⣄⠀⠀⠀⠀⠀⠸⣿⣻⣿⡿⠟⠛⢿⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠼⠿⡏⠀⠀⣻⠿⠆⠀⠀⠀⠀⠀⠀⠀⠘⢻⠟⠀⠈⢿⠋⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠏⠈⢻⣿⣦⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠉⠋


⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⢀⡉⠙⠳⣄⠀⢀⡴⠛⠉⣡⠀⠀⠀⠀⠀⢻⣿⣿⣳⣶⣶⣻⣿⣿⡇⠀⠀⠀⠀⣾⣿⣷⣤⣤⣤⣤⣾⣿⣷⠀⠀⠀⢸⣿⣿⣿⣆⠀⣴⣿⣿⣿⡆
⠀⠀⠀⠈⠛⠛⠛⣻⠛⢻⡟⠛⠛⠛⠀⠀⠀⠀⠀⠘⢿⣿⡗⠀⠐⣿⣿⠿⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⢿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠹⣿⣆⠀⠀⠀⠀⠐⣿⣿⣿⡿⠀⠘⣿⣿⣿⡿⠀⠀⠀⠈⠛⣿⣿⣿⠁⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⢾⣿⠃⠀⠀⠀⠀⠹⣿⠆⠀⠀⠀⠀⠘⣿⡿⠁⠀⠀⠘⣿⡿⠁⠀⠀⠀⠀⢸⢿⣿⣿⠀⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⢀⣠⠤⠀⠀⡀⠀⠀⢠⣄⡀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⢀⣴⣄⡀⠀⢀⣤⣦⡀
⠀⠀⠀⠀⢠⣾⣶⣄⠀⠀⣰⣿⣧⡀⠀⠀⠀⠈⠲⠦⠤⢼⣿⠶⠶⣿⠤⠤⠶⠊⠀⠀⢀⣿⣿⡿⣷⠀⢰⡿⣿⣿⣇⠀⠀⠀⠀⣿⣿⣿⣷⣀⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠈⠣⠀⠀⠀⠀⠀⢸⣿⣿⣯⣛⠿⢟⡫⣿⣿⣿⠀⠀⠀⠈⢿⣿⣿⣿⣹⣿⣿⣿⡿
⠀⠀⠀⠀⠘⠿⣿⣿⡟⢻⣿⣿⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆⣾⣿⣿⣿⠏⠀⠀⠀⣠⣾⣿⣿⡿⠀⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣴⣿⡟⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⣿⣿⣿⣿⠆⠀⠀⠘⠛⢿⣿⡿⠁⠀⠘⢿⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⣿⠿⠋⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣷⣷⡀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣻⣿⣟⣤⡀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⢄⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⣿⣿⣿⠉⢷⡀⠀⠀⠀⠀⢠⡿⢻⣿⣿⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣿⢿⣿⣤⠾⠃⠀⠀⠀⠀⢿⣅⣸⣿⣿⣇⣸⡗
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣽⣿⠆⠀⠀⠀⠀⠀⠀⠀⠉⣽⣳⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀
⠀⠀⠀⠀⠀⠀⢀⠼⠀⠀⢸⢤⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⡀
⠀⠀⠀⠀⠀⠀⡅⡀⠈⠈⠁⡀⡇⠀⠀⠀⠀⠀⠀⠃⠘⠀⠒⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣀⠃⠀⠚⣀⢣⠀⠀⠀⠀⠀⢀⠎⠡⣀⡠⠏⢢⡀
⠀⠀⠀⠀⠀⠀⡇⢃⠀⠀⢀⠇⡇⠀⠀⠀⠀⠀⠸⠀⡀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡊⢰⣹⠀⠀⠀⣎⡆⢣⠀⠀⠀⠀⠎⡜⡃⠀⠀⡘⢆⢡
⠀⠀⠀⠀⠀⠀⠘⣌⡂⠀⣊⡜⠀⠀⠀⠀⠀⠀⠀⢂⠱⡀⡰⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢬⢩⣒⡉⡤⠐⠁⠀⠀⠀⠀⠢⢌⡓⡤⠔⣋⠡⠌
⠀⠀⠀⠀⠀⠀⠀⢈⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡌⠁⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⠉⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⢤⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠘⢤⢠⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠼⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⢨⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡋⠉⠉⠙⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠙⢦⣄⡀⠀⠀⠀⠉⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⣀⡤⠖⠊⠁⠄⠈⠁⠄⣇
⠀⠀⠀⠀⠀⠀⣼⠁⢠⠔⡄⠀⠀⠉⠛⠲⠤⣄⣀⡀⠈⠳⣄⡀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣠⣋⡤⠤⠖⠒⠛⠉⠉⠀⣀⣈⠀⠀⠄⣀⡀⠀⢿
⠀⠀⠀⠀⠀⢰⡇⠠⠈⠀⠁⠀⠀⡤⡄⠀⠀⠀⠈⠉⠛⠓⠒⠛⠺⡏⠁⠀⠉⢟⠛⠏⠹⣯⠀⠀⠀⠀⣀⠅⡀⠂⠀⠃⠊⠀⠀⠀⠓⠃⠀⢸⡀
⠀⠀⠀⠀⠀⢸⠀⠀⣀⠀⠀⠀⠀⠑⠁⠀⡠⢄⠀⠀⢠⡒⡄⠀⠀⢹⡀⠀⠀⠈⠀⠀⠀⣻⠀⣠⠤⡀⠑⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢸⠀⠘⠔⠀⠀⣠⢄⠀⠐⠀⠉⠁⠀⢀⠀⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠈⠈⠀⠀⢀⡀⠀⠀⠀⢀⡤⢀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠳⠊⠀⠠⠀⠀⣠⡀⠀⠀⣖⡡⠀⢀⣴⠷⣤⣴⠴⠓⢺⠟⠉⡒⢤⣀⠀⠀⠃⠂⠀⠀⠀⠈⠐⠁⠀⠀⠪⠇⢠⡇
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠠⠂⠀⠠⠀⢀⡴⠋⣽⠀⠀⠀⠀⢠⠏⠀⠘⠋⠀⡏⠉⠒⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠂⠀⢆⠇⠀⠀⠀⠀⠀⢀⣤⢾⡟⠠⠀⡏⠀⠀⠀⢠⡏⠀⢠⢢⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠐⠀⠀⠀⠀⢐⣠⡶⠛⢡⠏⣀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠤⠴⠖⠋⠁⠀⢠⡟⠐⠔⠀⠀⣧⠀⠀⠀⢸⠀⢞⠆⠀⠈⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠂⠀⠀⠀⢻⡄⠀⠀⢸⡆⠀⠀⠈⠀⠂⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡀⠀⠐⠀⡀⢻⡄⠀⠀⢿⠀⠀⢌⡩⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠊⠸⠀⠀⡠⠄⠀⢻⠀⠀⠘⡇⢀⠀⠀⠀⠠⣲⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠀⠀⠑⠁⢈⡼⠀⠀⠀⢷⠀⠀⢀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣀⣠⠴⠚⠉⠀⠀⠀⠀⢸⡀⠀⠧⠃⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢂⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡠⠃⠀⠔⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠹⠑⠤⢤⣀⣣⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠣⡀⠀⠀⠄⢠⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠓⠊⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠒⠚⠁
⠀⠀⠀⠀⠀⣀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⡀
⠀⠀⠀⢀⠜⠁⠑⠮⡲⢄⣀⡀⠀⣀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⡀⠀⣀⣀⠴⡪⠖⠁⠘⢄
⠀⠀⡰⠃⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠀⠀⠀⠀⠀⠳⡀
⠀⡜⢇⠀⠀⠀⢀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⠟⡄
⢸⠣⣀⠑⢄⡠⠃⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⢢⣀⠔⢁⡠⢳
⢧⠴⠊⠑⠚⠓⢶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠓⠒⠉⠲⢤
⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈
⡧⠤⠤⠄⠐⠒⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠐⠂⠀⠤⠀⠤
⣷⣶⣶⣶⣶⣶⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣴⡶⠿⠛⠛⠿⠷⠶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠶⠿⠟⠛⠻⠷⣶⣄
⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣴⠶⠿⠿⠶⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⡀⢰⣿⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⣿⠀⠀⠀⠀⠀⠀⣿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⣀⠀⢀⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠉⠛⢻⡟⠋⠉⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠈⠛⠿⠶⢶⣶⡶⠶⠾⠟⠛⠛⠉⢁⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡌⠙⠛⠛⠿⠶⠶⣶⣶⠶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⢠⣾⣿⣅⣀⣀⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⢠⣦⣤⣤⣤⣤⣤⣤⣿⣷⡆
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢹⣇⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠻⣿⣿⡿⢿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣤⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢨⣿⠀⢰⡄⠀⠀⣰⠖⠀⠀⠀⢀⠀⠀⠀⣿⠂⠀⣾⡇⠀⢲⡀⠀⢸⣟⠀⢰⣦⠀⢠⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣯⠀⣸⡇⠀⢰⣿⠀⣸⡇⠀⢸⣧⠀⢰⡿⠀⠀⣿⡇⣶⠟⠀⠀⢸⡇⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢻⡇⢀⠘⠿⠀⣿⠇⠀⠘⠿⠄⢹⡇⠀⠘⠿⠀⣾⡇⠀⠀⠻⠇⢿⡄⠀⠀⢸⡇⠀⠈⠿⠀⠘⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠈⠙⠀⠀⠈⠇⠀⠀⠀⠀⠀⠈⠀⠀⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⢿⡆




⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⣾⠇⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠀⢰⣦⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠠⠋⠉⠉⠙⠛⠛⠻⠶⠶⢶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢄⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠶⠶⠶⣤⣦⣤⣄⣀⣀⢀⣀⠄⠀⠀⠀⠀⢀⣠⣷⣶⡿⠿⠋⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⡝⠛⠛⢏⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠃⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣦⡀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠁⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣆⢀⠀⠀⡁⠀⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠙⢻⣿⣿⣧⡤⢈⣀⢀⠀⣀⡴⠠⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣄⠁⠀⣶⡎⡉⣿⣯⠙⣾⣾⣷⡾⠿⣷⣾⣧⡆⠀⢤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⣿⣿⣿⣷⣶⠾⣧⢴⠭⣍⣤⡟⢯⠙⣿⣿⣿⠌⣿⣅⣨⢎⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⣫⠛⢿⡟⣎⣷⠈⢆⡹⣅⢀⠱⠊⠉⣈⣝⣉⡴⠿⠐⣾⢸⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠄⠙⠲⢬⡆⢰⠀⠜⢿⣧⣠⡙⢾⡇⣄⣈⡀⣋⢍⢸⣿⣶⣅⣋⣽⣔⠧⠊⠙⣼⠀⠦⣄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠳⢴⠄⠙⠋⣙⠂⠂⠠⣌⣉⣳⠶⣟⣫⣭⣝⣿⠯⣼⣿⣿⣿⠿⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠄⠠⠄⢸⠋⠰⣦⣿⣿⣿⣷⡻⠿⢻⡏⣴⣿⣿⣿⣷⣏⣠⢁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠀⣄⢀⣀
⠀⠀⠀⠀⠰⠀⠀⣌⠀⠠⢿⣿⣿⣿⡿⠳⠖⢸⡃⢿⣿⣿⣿⣿⡿⢽⣄⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⠟⠁⠀⠀⣿⠈⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠈⡀⠈⠀⠛⣔⣂⣉⢻⣜⠙⠾⣉⣤⣂⣰⣿⣿⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣼⠏⠀⠀⣠⠖⠀⠉⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠁⠃⠀⠙⠓⠚⢿⣉⠁⡨⠉⡙⠲⠛⢩⣯⣄⠈⠙⠻⢿⡆⠀⠀⠀⠀⣶⡴⠞⠛⠛⠉⡄⠀⠾⠏⠀⠀⠀⣠⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⡉⠉⠁⠚⢿⣧⣤⠞⠃⠀⠀⠀⠉⠀⠀⠀⠀⢄⠻⣤⡀⣀⣰⣟⢁⣙⣉⠀⠀⠁⠘⠋⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⣶⡎⠀⢀⡄⠀⠀⠀⢲⡄⠀⠈⠻⠙⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠉⠁⢀⣤⡄⠐⠐⠀⢀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠘⠦⣰⣿⣇⡀⠀⠀⠀⣷⡄⠀⠀⠀⣇⣼⣇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠀⢀⡀⣾⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠂⠀⠘⠋⠀⠀⠀⠀⠹⢿⣿⡲⡆⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣾⠁⠋⠀⠀⠨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⠶⠶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣦⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⡿⠀⢸⣿⡿⠉⢿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢸⣿⣿⠟⠁⠀⠘⠋⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠸⠛⠁⠀⠀⠀⠀⠠⠄⠀⢀⣸⡟
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠋⠛⣿⣿⣋⣀⣀⣀⣈⣉⣁⣀⣀⣤⣤⣤⣶⣶⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁⠀⠀⠘⠛⠻⠿⠿⠛⠛⠛⠛⠛⠛⠉⡉⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠈⠠⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠛⠉⠀⠀⢀⣤⣶⣾⣧⣀⠀⠀⠀⠀⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣦⣤⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢨⣤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠄⡳⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⡀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢺⣿⣿⣿⣿⣿⡿⠀⠹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠓⠆⠀⠀⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣿⡀⠀⠀⠀⣼⣶⣶⣶⣾⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⣿⣿⣿⠿⠷⠀⠀⠀⢿⣿⣿⡿⣿⣿⣿⣿⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⡀⠀⠠⡄⠀⠀⢰⡏⢸⡏⣀⢀⣀⡀⠀⢀⣤⢸⣗⡀⢰⡇⠀⠀⠀⣿⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢺⡇⡸⠇⣏⠾⢽⣧⣿⢻⣏⠼⣿⡁⣼⡇⢾⣼⠃⢿⠀⠂⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠉⠀⠈⠁⠀⠈⠉⠀⠀⠛⠙⠷⠀⠈⠋⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠆⠈⠉⠰⠄⠀⢰⠂⠈⠉⡏⠁⠋⠁⠁⠁⠀⠁⠋⠀⠀⠈⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡦⠀⣀⣠⠀⢀⠀⡀⠀⠀⡇⠀⡄⠒⠀⡄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠧⠠⠄⠀⠄⠌⠀⠀⠄⠀⠀⠀⠀⠄⠄⠥⠀⠀⠀⠦⠄⠀⠠⠄⠠
⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡄⣤⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⡀⢈⠀⠃⠃⠛⠃⠋⡁⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣙⠋⠀⡐⢐⣒⢠⡀⠙⠉⠀⠍⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠒⠒⠁⠀⠀⠀⠀⢠⣟⡅⠀⣼⠋⠀⠀⠈⢆⠈⢓⣵⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠈⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢀⠄⠹⣟⠠⣀⣿⡎⢰⡖⢰⣧⣰⡆⠈⣴⠂⢠⡖⣿⠀⠀⠠⡀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡏⠸⣇⠜⠗⠃⣼⡇⣿⠁⠙⠖⠉⠉⢻⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠹⣿⣿⣿⣷⣸⣷⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠛⠉⠉⠉⠛⢿⣿⣿⣆
⠀⠀⠀⢀⣶⣶⣤⣤⣬⣭⣉⣉⣛⡛⠃⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⢀⣠⣤⡀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠈⠙⠛⠋⠛⠱⠿⠿⠿⣿⠟⠀⠀⠀⠀⠀⢰⣦⣤⠀⠀⢀⣙⣛⣿⣿⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠛⠿⠟⠁⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⡀⠀⠀⠀⣀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢦⣄⢠⣦⠀⠀⣶⠤⠄⢀⣴⡄⠀⣤⡶⠄⣴⡶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡿⢸⣿⠀⠀⣿⣤⠀⢸⠟⣧⠈⢿⣤⡀⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⢸⣿⣀⠀⣿⣀⡀⣿⠛⣿⠀⣀⣹⡇⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠀⠉⠉⠁⠉⠀⠈⠁⠉⠁⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡄⢠⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⡃⢸⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣸⡷⢸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⢀⣤⡄⠀⣤⣤⣄⠀⣤⣤⠄⣤⣤⡄⢠⡄⢠⡄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣸⠿⣇⠀⣿⣠⡿⠀⣿⣤⠀⣿⣄⡀⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⡀⣿⠶⣿⠀⣿⠙⣷⠀⣿⣀⡀⣿⡉⠁⢸⣇⣸⡇⢸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠀⠉⠁⠉⠀⠉⠀⠉⠉⠁⠉⠀⠀⠀⠉⠉⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⡦
⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⣀⣀⣀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣠⣀⣀⣀⣀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠤⣴⡟⣚⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠀⢤⡀⠀⠀⣰⡟⣰⢿⡾⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⡄⣸⢇⣠⣀⠀⠀⠀⣀⣁⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⢡⣾⠵⠋⠀⠀⡎⢀⡾⠀⢠⡟⣸⢀⣴⢋⡞⠀⠀⣠⠞⣱
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢰⡿⠓⢲⡿⠁⠀⠁⠀⠀⠳⢏⠀⣠⣿⠔⢁⣾⠧⠊⠀⠀⣴⠯⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢠⡿⠁⢠⡿⣱⣆⣴⢂⡖⣰⠆⣤⢠⡿⣳⢀⣾⠁⣴⢠⡖⣰⠏
⠀⠀⠀⠀⠀⠸⠀⠄⠀⠀⠻⠇⠐⠟⠘⠋⠺⠇⠿⠐⠛⠸⠃⠟⠻⢁⡾⢻⠘⠃⠛⠐⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠔⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⢟⣭⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡦⠊⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢙⣷⣿⣿⣿⣎⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠛⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⣠⣾⣿⣿⣿⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡖⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢈⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠐⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠢⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⢿⣿⡿⣣⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⢿⢿⠇⠀⢰⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡿⠿⢿⠀⠀⠾⠇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣇⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣶⣦⣾⣧⣴⣶⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣼⣿⢿⣿⡏⣵⣟⣿⢫⣽⣿⣿⣇⣠
⠀⠀⠀⠀⠀⠀⠀⢾⣾⣿⣿⣧⡻⣿⣷⣌⣙⣻⣬⣙⣿⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡿⣫⣽⡝⣿⣿⣿⡿⠿⣿⣻⡿⣾⣿⣿⣤⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⢁⣿⠳⢟⣫⣼⣿⣿⣿⢻⣇⡿⣻⣿⣿⠏⣾⣷⡦⠀⠀⠀⠀⣰⣿⣿⣿⣿⣦⣤⣄
⠀⠀⠀⠀⢰⣾⣿⣟⢰⡇⢸⣻⡿⢿⡏⢻⢻⠀⡛⡇⣹⢿⢿⠀⣿⣿⣶⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⢠⣾⣿⡇⢸⡇⢾⠸⢸⡟⡇⢸⠘⢠⣹⡃⢫⣘⣘⡃⣿⡿⣿⣿⣿⣷⣦⡘⣿⣿⣿⣿⡿⠉⣸⡇
⠀⠀⠀⠀⠀⣽⣿⣧⢸⣇⣼⣿⣦⣦⣽⣿⣾⢟⣿⡇⢻⣿⠏⣿⣮⣿⣛⣿⣿⡟⣭⣷⣸⣿⣿⣿⡁⠀⣿⣇
⠀⠀⠀⠀⠀⠉⣽⣿⣌⣻⣿⣿⣿⣿⢋⣵⣿⡼⢿⡿⢖⣿⢠⣿⣿⡇⣹⣿⣼⣇⢹⣿⢟⣻⣿⣿⣇⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⡘⢿⢹⣿⢃⡇⠿⢻⢸⡇⣶⡅⣭⣇⢿⠸⡌⢿⣶⣯⣿⣿⣿⣆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⢀⣠⣿⢟⣯⢰⡿⢨⣵⣿⣶⣬⣷⣬⡇⢿⣿⢖⣟⢻⣷⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣿⣿⣿⡿⡿⢃⡾⢸⢇⣿⢡⣦⡿⢛⡟⣻⣬⣿⣜⣿⠘⣟⣿⣿⣿⣿⣻⣿⣇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡾⡏⢰⢟⢿⡘⣇⢺⠃⣶⡟⣽⢸⣿⣿⣿⣿⣭⣾⣿⣿
⠀⠀⠀⠀⠀⢨⣶⣿⣿⣿⢡⣶⣿⢿⣿⣿⡇⣼⠟⡄⣩⠸⣟⣷⢸⡇⣷⣽⣤⣿⡌⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣛⣿⣿⣿⣮⣟⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣋⣸⣵⣿⡿⢛⣿⣿⣿⡏⣾⢿⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢯⣶⣿⣷⣶⣬⣭⣙⣛⠻⠿⠿⠋⠴⠿⢟⣱⣿⣿⣶⣾⠟⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣮⣯⣿⣿⣿⡿⣯⣭⣭⠝⣃⣐⠶⠿⠿⢿⣿⠿⠿⢛⣡⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣭⣿⣿⣿⡿⠟⠛⠻⢿⣿⣦⣤⣴⣾⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠈⠉⠉⠁⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⣿⣇⠀⠀⢀⣤⡾⢿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠳⠀⠈⠀⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⠟⡘
⠀⠀⠀⡀⠀⠀⠢⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠘
⠀⠀⠀⠹⢶⣤⣤⣀⣈⣉⣙⡛⠛⠿⠿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⣿⣿⣿⣿⣶⣤⣈⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⣀⣀⣀⣈⣉⣙⡛⠷⡄⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⢦⠀⣿⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠿⠿⠟⠛⢛⣒⠀⠸⡆⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠶⠾⠛⠋⣉⡭⠄⠈⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠁⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠠⢤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⡆⢠⠠⡀⢤⡀⢤⣌⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⢰⠇⢸⡇⢻⡆⢻⣦⠹⣿⣦⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠸⠀⢸⡇⢸⣿⠘⣿⣇⠹⣿⣷⡘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠀⢻⣿⡀⠹⣿⣷⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⡄⠘⣿⡇⠀⢻⣿⣧⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⣿⡀⠀⠻⣿⣆⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠄⠀⠙⢿⣦⣀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⣰⣿⣿⣿⡶⣷⣿⠁
⠀⢠⣾⣿⣿⣿⣿⣾⣫⣾⣇
⢴⣿⣿⣿⣿⣵⣟⣽⣿⣿⣿⠆
⠀⠀⠉⣩⡾⢫⣿⣿⣿⡿⠋
⠀⠀⠀⠙⠁⠘⣿⣿⠟⠁
⣤⣤⣤⣤⣤⣤⣬⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣥⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣾⠇⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣼⡟⠀⢠⠊⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣿⠁⡰⠁⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠀⠀⢰⣿⠀⠀⠀⣸⡿⠊⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⢸⡟⠀⠀⠠⣿⡇⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⡔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⣾⡧⠀⠁⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⣿⡇⠌⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣀⣶⡀⢠⡞⣡⡄⠐⢹⡖⣽⡇⢀⣿⡟⣰⠟⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡏⣿⡇⣿⣃⣿⠇⠂⣿⠁⣿⠁⣼⣿⣿⡅⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠉⣿⠁⢻⡷⠀⠁⠻⠎⠀⠉⠀⣿⢰⢹⣿⢻⣇⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠜⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠛⠃⠸⠿⠈⠻⣄⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢳⣦⠀⣰⡄⢾⠀⣴⡌⡆⠀⣠⣶⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⣿⡀⣹⣇⠈⢲⣿⠄⢼⠀⣿⡇⢠⣿⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⣿⡇⡟⣿⡄⠹⣿⠀⡆⠀⣿⠇⣸⣿⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢁⣿⣇⠀⠙⠊⠀⠀⠈⠛⠈⣿⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡼⠁



⣶⡶⢶⢶⠶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⡶⢶⢶⣶⣶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢦⣀⢄⡴⠢⠞⠁⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡄⠀⢀⡀⠤⠒⠉⠉⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠈⡀⠀⠀⠀⠀⢀⡼⣡⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⣆⠀⢸⠁⢀⣠⢤⡴⢏⡤⠶⠶⢤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡎⠀⢀⠔⠚⢤⡎⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠒⢦⣴⣿⠃⠀⡇⠀⠀⠀⡗⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⠙⠛⠀⠉⠙⠳⢤⣈⠢⠤⡿⠳⢖⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢲⣆⠀⠀⠀⠀⠉⢻⠶⠶⠶⠶⠟⠿⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⣧⡀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠘⢷⡤⠴⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⡜⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠓⢇⡀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⠀⠉⠲⢄⡀⠀⠀⠀⠙⡦⠤⠴⠒⠋⠉⠔⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠑⠦⡀⠀⠉⠲⡄⠀⡼⡁⠀⠀⠀⠀⠀⠠⠊⢁⡠⠤⡔⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠀⠀⠀⠀⠘⣷⠖⠀⢱⡞⣷⢷⠀⠀⠀⠀⠀⠀⠰⣅⠀⢀⣇⣈⡏⣦⠔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠎⠀⠀⠀⠀⢈⣿⣀⣀⣀⢁⣿⠀⠳⡀⠀⠀⠀⠀⢀⠟⢒⣉⣀⣀⣿⡥⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣔⡭⣧⡜⠿⢀⡀⠀⠀⡟⠉⠀⠀⠀⠀⠀⢸⣤⣴⠟⠀⢀⡠⢚⡡⠒⠐⢷⠞⠋⠀⣳
⠀⠀⠀⠀⠀⠀⠀⢸⢶⡡⣿⣇⣄⣀⡟⠻⡋⠁⠀⠀⠀⠀⠀⠀⢸⠛⣄⠳⠒⣁⠔⠉⠀⠀⠀⠈⠓⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠸⠬⠂⡝⠀⠈⠉⡷⢤⣀⡀⠂⠄⣀⢀⣀⣀⣸⣀⣈⣳⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠓⠦⢄⡉⠣⢤⠴⠋⢻⣀⣠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠶⠶⠶⢶⠒⠚⢳⠀⠀⠀⢸⢛⡏⠉⠉⠉⠉⠑⠒⠒⠒⠊⠉⠉⠉⠉⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢈⢉⣯⠿⣤⠴⣤⣼⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠛⠛⠋⠉⠛⠛⢛⣉⣉⠀⠀⠄⠀⡈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠄⠶⢮⣥⣠⠴⢆⡀⠀⠀⠀⠘⠀⠂⠈⠀⠤⠤⠄⠀⠁⠀⠁⠊⠀⣀⡑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠠⠤⠤⠤⠤⠤⠤⠤⠤⠔⠖⠒⠒⠒⠒⠒⠒⠂⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣠⠤⣄⡀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢴⡷⣼⠕⠁⠀⠀⠀⠠⠀⠈⠑⢲⡿⠐⠀⢠⠞⡁⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡇⢄⠀⢸⡇⢿⡐⣶⠀⡴⢛⡀⣷⠎⡿⢸⡅⡸⢃⣟⠀⣴⢀⡄⣴⢃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠊⠀⢸⡇⠸⡇⢿⡐⠃⠺⠆⠻⠐⠧⠘⠛⠇⢸⡗⠈⠧⢾⣱⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠼⠁⠀⠀⠀⠀⠀⠀⢠⣸⠂⠀⠀⠀⠀⠸⣇⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢅⡐⠈⠉⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠓⠔⠛⠋⠀⠀⣠⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠉⣄⠀⠰⣄⡀⣸⠧⠖⠒⢶⡇⢀⠀⡄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣃⡿⠀⣹⠆⣴⢛⠂⣾⠀⢰⡿⢸⡇⠸⣄⢧⡹⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⠞⠁⠀⠻⠀⠁⠛⠀⢿⡠⠙⠒⠸⡏⢂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣤⣤⣤⠀⣀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡆⠀⣶⡚⠳⣦⠀⢸⢿⡆⠸⠘⣇⠺⠀⣿⠰⠇⢸⡇⢸⠁⣿⠀⣶⠄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠓⠀⢸⣇⣠⠟⠀⢸⡨⣷⠄⠀⣿⠀⢀⡿⠂⡄⣾⠧⠀⢸⡇⢰⡟⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢲⡄⣿⠙⠷⠄⠚⠀⠘⠃⠀⢋⡀⠈⠛⠚⠁⠛⠀⠀⠹⣧⡾⠃⣼⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠃⠀⠀⠀⠀⠶⠀⠀⠀⠀⡿⢿⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⠙⠒⠃
⠀⠀⠀⠀⢸⡟⢻⡟⢻⡇⢤⡄⠀⣤⠄⠀⠀⣤⡄⠀⠠⣤⡀⢤⡄⠠⣤⠀⢠⡄⠀⣤⡴⢤⡄⣤⡄⢠⣤⠀⣤⡄
⠀⠀⠀⠀⠰⠃⢸⡇⠘⠆⢸⡇⠀⣿⠀⠀⢸⢿⣧⠀⠀⣿⣧⢸⡇⠀⣿⢀⡟⠀⠀⣿⡇⢸⡄⣿⡇⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣧⣤⣿⡤⠀⣾⠈⣿⣀⠀⡇⢻⣿⡇⠀⣿⣾⡀⠀⠀⣿⣇⡀⠀⣿⡇⢸⣿⠀⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⣿⠀⢰⡏⠉⢿⡏⠀⡇⠈⣿⡇⠀⣿⠉⢷⡄⠀⣿⡇⠀⠀⢻⣇⣸⡿⠀⣿⡇⣠⡇
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⣁⣀⠉⠀⣨⠄⠀⠈⠁⠀⠁⠀⠈⠁⢀⣉⡠⠬⠻⣶⠛⠑⣦⠀⣀⣈⠁⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠞⠛⠉⠈⠁⠸⣿⣄⣀⣀⣀⣤⣤⠶⠖⠛⠉⠀⠀⠀⠀⠈⠛⠒⠋⠈⠀⠉⠛⠲⠶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠲⣤⠀⢲⡎⠉⠛⢻⡍⢳⠆⣠⡖⢲⡆⢠⡶⢲⡆⠐⣶⠒⣶⠀⣶⠒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡿⠀⢸⡇⠀⠀⢸⡇⠸⠆⠻⣧⠸⠆⠸⣧⡘⠇⠀⣿⠀⠷⠀⣿⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⢷⡄⢸⡇⠀⠀⢸⣇⡀⠀⠀⠘⢷⡄⠀⠈⢻⣆⠀⣿⣠⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⠇⢸⡇⢀⡇⢸⡇⢀⡆⣾⠁⢸⡿⢸⡇⢀⣿⠀⣿⠀⣸⠀⣿⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉⠉⠁⠉⠉⠉⠁⠈⠙⠋⠀⠈⠙⠋⠁⠀⠉⠉⠉⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣾⡿⠞⣡⡞⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢻⣿⣟⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⢠⣄⢈⣟⠚⠁⠉⣡⣶⣦⠀⢀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢹⡆⠾⠻⣷⢸⣿⡟⢹⣆⣤⡄⣿⡇⣿⣄⠛⣛⣿⡿⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢨⣥⣤⣤⣿⢸⣿⠃⣬⣿⣡⡿⣿⡷⢴⡟⢸⣿⣿⡇⡀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣾⢿⣤⣾⢿⣾⣿⠀⠙⢿⣭⡄⣿⡇⢸⣇⣼⣿⣘⠻⠃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⣬⠀⢀⢀⣄⠀⣀⣀⣩⣴⢾⡀⢀⡙⢁⣼⣟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡗⢻⡿⠖⠘⠸⣟⠃⠛⠛⣿⢻⣏⣰⡟⢻⣏⣿⠙⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⣼⡇⣤⠀⣶⢸⡇⠀⠀⣿⡏⣿⢻⡇⢸⣿⣿⢰⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠿⠏⠀⠙⠛⠁⠀⠀⠙⠇⠸⣯⣿⠟⠉⠛⠿⠁⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉




⣿⣟⣛⣟⣛⣻⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠉⠉⠉⠉⠙⠲⢄⣿
⠀⠀⠀⠀⠀⠀⣴⣟⠁⠀⠄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢶⠄
⠀⠀⠀⠀⠀⣸⣿⠨⠁⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢲⡆⢰⢀⡤⠒⠒⠲⣤
⠀⠀⠀⠀⠀⣿⣿⠀⡀⠀⠀⠀⠠⠄⠐⠒⠒⠒⢲⣶⠀⣿⠀⠸⣿⡄⣾⠃
⠀⠀⠀⠀⠀⢹⣿⣇⠡⢀⠀⠀⠀⠀⠀⠀⡆⢠⣿⡟⠀⣿⠀⠀⣿⠁⢻⣇⡀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠻⣿⣄⡈⠀⢌⢎⡦⠠⢴⢅⣼⠟⠀⠀⠉⠁⠀⠈⠀⠀⠈⠉⠉⠁
⠀⠀⠀⡆⠀⠀⢀⠌⠛⠿⠷⠾⠋⡃⡂⠋⡛⠀⠀⠄⢰⠀⣶⣶⠂⠂
⠀⠀⠀⠰⠀⠀⠈⢀⣤⡠⡤⣦⢁⠠⡀⠡⠃⠀⠀⠀⠘⠀⣿⣿⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄
⠀⠀⠀⠤⡄⣠⠀⣬⣸⠃⠀⠈⢉⢀⠩⠁⢆⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⠀⢸⡇
⠀⠀⠀⠀⡌⠡⠁⠀⠀⠀⠀⠀⠺⠆⠉⣦⡾⢠⠬⣅⠀⠀⣿⣿⠀⠀⣿⡏⣿⡆⠀⣸⡄⠀⣿⡆⣿⡏⣿⡇⢸⣇⡔⠁⢰⣯⠐
⠀⠀⣆⠸⠧⢄⠀⠀⠀⠀⠀⠀⠄⠀⠐⠟⠹⠇⠀⠈⠀⠀⣿⣿⠀⠀⣿⡇⣿⣇⣾⠁⠀⠀⣿⡇⣿⡇⣿⡇⢸⡏⣿⡄⠀⠙⣷
⣀⡀⠄⠀⠀⡀⠠⣆⡄⠀⠀⠈⠰⡖⠁⠀⠀⠀⠤⠆⠀⣀⣿⣿⠀⠀⣿⡇⣿⡏⠿⣦⣀⡀⣿⡇⣿⡇⣿⡇⢸⡇⠘⣷⡀⡄⣸
⠀⠀⣠⢀⠈⠐⢰⠂⠀⠀⠀⠀⣦⡂⠀⠐⠁⠀⠐⠀⢀⠄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀
⠀⠀⢱⠟⡅⠐⠒⠀⠀⠐⣄⣠⠉⠈⢡⣗⠆⣠⣄⢈⣥⠀⠀⡰⢄⣄⠠⠤⠤⠀⢀⠀⣀⣀⣄⢀⣰⣿⣟⣧⣄⣐⣦⣶⡤⠔⠂
⠀⠀⠀⠑⠂⠲⣒⠚⣫⠀⠀⠀⠀⠀⢠⡁⣆⢉⣼⠏⣀⠀⡰⠃⠈⠀⠹⣿⡿⠟⣒⣶⣖⣍⣍⣛⣩⣥⣬⣋⠈⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠋⠦⠮⢰⡀⢰⣟⣁⡹⢤⣤⣶⠾⠄⣍⠳⣮⡙⠉⢛⠿⠞⠧⣥⠎⢽⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢼⡦⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣇⠀⠀⡀⠋⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⣀⡂
⠀⡴⠴⢑⢠⠀⡊⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣩⣭⣭⣍⡛⢿⣦
⠀⢀⣁⣐⠈⢀⡇⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢡⣿⠋⣥⡌⠹⣷⠘⣿⡇
⠀⠀⠘⢃⣀⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⢿⣤⣉⣡⡼⠏⣰⣿⢁
⠀⠀⠲⠬⠗⠚⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣬⣭⣭⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⣩⡅
⠀⠀⢰⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢿⡇⢹
⠀⠀⡜⠀⠀⠀⠀⢈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠳⣄⠀⠀⠀⠀⠀⠀⣸⠁⢸⡇⢸⡇
⠀⢸⠀⠀⣀⣀⣠⠆⠀⠀⠉⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢎⠀⠀⠀⠀⠀⠈⢠⡏⠀⢸⡇⠀⣧
⠀⠀⠓⠦⠬⠥⠥⠤⢄⣀⣀⣀⡖⠁⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠃⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣤⣤⣌⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⢠⡿⣫⣭⣻⣧⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⢀⣿⣻⢋⢹⡇⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠄⠀⠰⡴⠋
⢸⡿⣏⠿⢻⣧⡟⠀⡋⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡘⡠
⢸⣧⠻⠶⢟⡿⠁⠸⡇⠀⠀⠀⠀⠈⠂⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶
⠀⢻⣶⣶⣿⡁⢰⡇⢻⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠘⠷⢬⣥⡞⢻⠙⡇⠸⡄⠀⠀⠀⠀⠀⢰⠸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⢰⠁⠸⢰⡇⠀⣇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀
⠀⠀⠀⡞⠀⠀⢸⡇⠀⢻⠀⠀⠀⠀⠀⠀⠣⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⠃⠀⠀⢸⡇⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⡟⠀⠀⠀⣼⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⡈⢆⠀⠀⠀⡀⠀⠀⠀⠀⡄
⠀⠘⠁⠀⠠⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠘⠁⠀⠀⠀⠀⠁⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡲⢠⠀⠀⢊⠐⠶⠄⠀⠀⠠⡶⠀⡁⠂⠀⡄⢒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢖⠐⢀⣠⠗⠢⡅⠒⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠒⠨⠔⠺⣄⡀⠂⡰⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋⣠⠖⡃⠐⠀⠀⠀⠀⢀⣤⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠂⢘⠢⣄⠙⠦⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡴⠷⠒⠋⠁⠀⢀⡀⢀⣀⠀⢰⡿⠁⠀⢛⣃⡀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠈⠙⠒⠾⢧⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠋⠀⠀⠀⠀⠀⠀⢹⣿⢸⣿⣧⣿⡇⠀⠀⣿⣿⣇⢸⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡃⠒⠀⠀⠀⠀⠀⢸⣟⢸⣿⢿⣿⡇⠀⢰⣿⣾⣿⢸⣿⣄⣀⢸⣿⣀⣀⠀⠀⠀⠀⠀⠒⢘⠊⢧
⠀⠀⠰⢀⡂⣠⠀⣼⢁⡞⡁⣃⠀⠀⠀⠀⠀⠘⠋⠘⠛⠈⠛⠃⠀⠘⠛⠀⢿⣎⢛⣛⠛⠘⠛⠛⠋⠀⠀⠀⠀⠀⡘⢈⢳⡈⣧⠀⣄⢐⡀⠆
⠀⠐⢨⢩⠑⢠⢰⠇⡾⣠⠱⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠈⠻⠾⠟⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠡⠌⣄⢷⠸⡆⡤⠊⡍⡅⠂
⠀⠀⠀⢸⢰⡆⡾⢠⣷⠋⢃⣀⣀⡀⣀⡀⣀⡀⣀⠀⣀⡀⢠⡿⠁⣐⡟⠀⣀⣀⠀⠀⠀⣀⣘⠟⣹⣧⡀⣀⡀⣀⣀⡘⠘⣾⡄⣷⢰⡆⡇
⠀⠀⠀⠸⣾⡇⣇⡼⠁⠀⠛⢻⡟⠃⣿⡇⣿⡇⣿⡇⣿⣧⣾⡇⣾⡟⠿⠐⣿⣿⠇⠀⣾⡟⠿⠀⣿⢻⣧⣿⠃⣿⣏⡉⠀⠈⢧⣸⢸⣿⠇
⠀⠀⠀⠀⠹⣷⡏⠀⠀⠀⠀⣸⡇⠀⣿⡟⣿⡇⣿⡇⣿⡿⣿⡇⢿⣧⣾⡶⣮⣿⡇⠀⢿⣧⣼⡆⣿⠀⣿⣿⠀⣿⣯⣥⠀⠀⠀⢹⣾⠏
⠀⠀⠀⠀⠀⣧⡧⢄⠀⠀⠀⠈⠁⠀⠈⠁⢹⣇⢩⡄⠉⠀⠉⠁⠀⠉⠉⠀⠈⠁⠀⢀⠀⠈⠉⠁⠉⠀⠉⠁⠀⠉⠉⠁⠀⠀⡠⢼⣼
⠀⠀⠀⠀⠀⠸⣷⢨⡕⠰⢐⠂⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠆⢪⡅⣼⠇
⠀⠀⠀⠀⢀⡠⠼⡆⣧⢹⠰⠀⠀⠀⢠⣶⣶⣶⢰⣶⢠⣶⠀⣴⣶⡄⢰⣦⠀⣿⢱⣶⣴⡶⣰⣶⣦⠀⠀⠀⠀⠀⠀⠖⡏⣼⢰⠧⢄⡀
⠀⠀⠀⠀⠀⠆⢧⣹⡸⣾⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⣼⣿⢀⣿⣻⡇⢸⣿⣿⣿⢸⣿⣿⠃⢘⣿⣏⠀⠀⠀⠀⠀⠀⠀⣷⢇⣏⡼⠰
⠀⠀⠀⠀⠀⠀⠀⢙⢳⡙⣆⠀⠀⡀⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⠻⣿⢸⡿⢻⣿⢸⣿⢻⣆⠻⣿⡿⠀⠀⠀⢀⠀⠀⣰⢋⡞⠋
⠀⠀⠀⠀⠀⠀⣒⠨⣔⠻⣼⡆⣤⡕⢄⢃⢂⡀⠀⠀⠀⣿⡤⣶⡀⠻⣦⣰⡆⠀⠀⠀⠀⠻⣦⣶⢀⡐⡚⣄⢪⣤⢰⣧⠟⣢⠅⣒
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠳⣌⢳⡀⠙⢆⡈⢳⡀⠀⠀⠀⠈⠛⠛⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⢀⡞⢁⡴⠋⢀⡞⣡⠞⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣈⠳⣄⠙⠶⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠴⠋⣠⠞⣁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⠽⠦⣄⡉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢉⣠⠴⠯⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠺⠧⢤⣤⣀⣀⣀⣀⣀⣀⣤⡤⠼⠗⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡏⠉⢻⣷⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣾⡟⠉⢹⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣿⡇⠀⢸⡿⠀⠀⢀⣴⣾⠿⠛⠛⠿⣷⣄⣠⣾⠿⠛⠛⠿⢷⣦⡀⠀⠀⢻⡇⠀⢸⣿⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⢠⣿⠃⠀⣾⡇⠀⢀⣾⠏⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⢀⣤⡀⠹⣷⡀⠀⢸⣿⠀⠘⣿⡄⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢸⣿⠀⢀⣿⠃⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠃⣠⣿⠃⠀⠈⣿⡄⠀⣿⡇⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣀⢸⣿⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⣿⡇⢀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣰⣿⠟⠿⣿⣿⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⣿⣿⠿⠻⢿⡆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⣷⡀⠀⠈⠻⣷⣦⡀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⢀⣴⡾⠟⠁⠀⢀⣾⡟⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠙⢿⣦⡀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠙⢿⣦⣴⡿⠋⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⣰⡿⠋⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠙⢿⣦⡀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠈⠻⠟⠁⠀⠀⢀⣤⣾⠟⠉⠀⠀⠀⢀⣾⠟⠁⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠈⠙⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⠋⠁⠀⠀⠀⠀⣴⡿⠋⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠘⠛⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⠛⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣧⣀⣀⡀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⢀⣀⣀⣼⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠉⠉⠛⠛⠿⠿⣶⣶⣾⣿⡀⠀⠀⠀⠀⢀⣽⣿⣶⡾⠿⠿⠛⠛⠉⠉⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⣿⡇⠀⠀⠀⠀⢸⣿⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢰⣿⠃⠀⠀⠀⠀⠘⣿⡆⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠏⠀⠀⠀⠀⠀⠀⠹⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠈⠈⠀⠀⠀⠐⠀⠂⠘⠀⠀⠀⠀⠃⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠐⠈⠀⠀⠠⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡌⠀⣀⠴⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠸⡄⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠞⠑⣄⢇⡀⠀⢀⡤⠇⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⠀⢀⠴⠋⠀⠈⢦⣉⣉⣁⣀⠤⠒⠉⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠈⡹⢍⠳⡀⠀⠀⠀⣀⣀⣇⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⣠⠔⠉⢫⡀⠀⠑⠈⢢⡠⠊⠁⠀⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢩⠃⢀⡎⠁⠑⠤⠤⠴⠋⠀⠀⠀⡴⠁⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡏⠈⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠓⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⡇⠀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠐⠢⠤⠤⠔⠂⠀⢠⠎⠀⠀⠀⣙⣚⡁⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠤⠀⠀⠀⡇⠀⠀⠀⠀⠧⠤⠇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁


⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣿⣿⣟⣿⣿

⠀⠀⠀⠀⠀⢀⡆⠀⠀⣠⠞⠁⠀⠀⠀⢀⡔⣿⢳⡄⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠸⣗⣾⣸⡟⠀⡀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⡀⠀⢰⡟
⠀⠀⠀⠀⠀⣾⡇⠀⣿⠁⢠⡄⠀⢀⡴⠀⠈⠉⠁⠀⠀⣿⠀⠀⣿⠀⢠⡀⠀⣾⡁⠀⣄⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠁⠀⣿⡀⢸⡇⠀⣾⠃⢸⡇⠀⣿⡄⢰⡿⠀⢀⣿⢶⡟⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⣼⡇
⠀⠀⠀⠀⠀⢹⡆⠀⠙⠀⢿⠁⠀⠙⠂⠸⡇⠀⠙⠃⢸⡇⠀⠀⠛⠸⣇⠀⠀⣿⠀⠀⠛⠀⠸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⡴⢒⣶⠶⣤⡀
⠀⠀⠀⠀⠀⣠⠏⣰⠟⢡⡭⣬⢷⡄
⠀⠀⠀⠀⣰⠏⣼⠇⡴⠁⠀⠈⢷⣿
⠀⠀⠀⢠⡟⢰⡯⡞⢠⠖⠒⢦⠘⣿⡇
⠀⠀⠀⣼⢀⡿⢸⢡⢯⡞⠙⢶⣧⢻⣷
⠀⠀⠀⡏⢸⡧⡞⣾⡟⣴⣿⡆⣿⢸⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⡤⢤⠤⣶⠂
⠀⠀⢸⡇⢸⠒⡇⣷⣿⡿⣿⣷⣿⣼⣿⡦⠤⠤⠤⠴⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠻⣿⢻⡟⠛⡻⣇
⠀⠀⢸⡇⢸⡽⡇⣿⡟⣟⡟⡏⣿⢸⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⡇⢸⡗⡇⣗⣇⠙⠚⣸⡏⣸⣿
⠀⠀⠀⢿⠘⣧⢻⠸⣎⠢⠴⡿⢡⢿⡇
⠀⠀⠀⢸⡇⢿⢪⣆⠈⠓⠋⢁⡎⣾⠁
⠀⠀⠀⠀⢳⡘⣇⠈⠢⣤⣴⣛⡿⠃
⠀⠀⠀⠀⠀⠳⣜⣷⣔⣊⣥⠟
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡟⠛⠋⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⣿⣄⡀⠀⠀⠀⣀⡀⢀⠀⣀⡀⠘⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠴⣿⠀⡇⠀⢠⡞⢹⡇⠸⣿⠉⣿⠀⣿⡼⠃⠀⠀⠀⠀⠀⠀⠘⣧⢰⡇⣴⠛⢶⣸⡇⠀⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⣿⠀⣇⣠⣾⣀⡿⣧⣀⡇⠀⣿⢀⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⣷⣀⡼⠻⣇⡼⣷⡀⢘⡁
⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠉⠁⠈⠉⠀⠈⠉⠁⠀⠈⠉⠁⠈⠁⠀⠀⠀⠀⠐⠀⠘⠀⣸⠇⠈⠉⠁⠀⠀⠀⠈
⠀⠀⢸⡙⠳⣯⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣤⣀⣠⡴⠏
⠀⠀⣠⠽⠦⣝⡾⠛⠛⠓⢦⡀
⠀⠀⠻⠤⠶⢺⡇⠐⠀⠒⠈⡇
⠀⠀⠀⠀⠀⠀⢧⠀⠘⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣷⣀
⠀⠀⠀⣰⠞⠉⡟⢧⡴⠋⠁⠀⢠⠞⠋⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⢼⣧⣟⣹⣇⠀⠀⠀⢐⡶⠶⢧⣀⠀⠀⠀⠀⣠⠿⠦⣤⠄⠀⠀⢠⠰⣇⡆
⠀⠀⠀⣇⠘⢀⠀⢸⡇⠀⠀⢠⡏⠀⠃⢀⠀⣽⠀⠀⠀⡾⠛⣆⢰⠇⡟⢸⡆⢹⡟⠒⠛⠀⠀⠘⡏⠰⠀⡌⣧⠀⠀⢀⣼⠃⠆⢠⠘⡷⠀⠀⠈⠻⡟⠀⣼⣡⠄
⠀⠀⠀⠘⢦⣀⣀⣼⣧⣤⡀⢸⠁⠐⠃⣠⠼⠁⠀⠀⠀⣇⠀⢹⣼⠀⣇⢸⡇⠈⡇⠀⡤⢤⠀⠀⠹⣤⡀⢠⣧⣤⣶⡀⣿⠀⠲⠀⣸⠃⠀⢀⠀⢀⡷⠚⠛⠳
⠀⠀⠀⠀⣠⠟⠁⢀⡀⠀⠙⢾⡤⠖⠛⠁⠀⠀⠀⠀⠀⠈⠳⣼⣿⠀⣿⢸⡅⢠⡇⢸⠃⠸⡇⠀⠀⢦⡽⠋⠀⢠⡀⠹⣿⢀⡤⠞⢁⠀⠀⣸⢧⣸⠁
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⡄⢸⡆⠀⠀⢀⡀⢠⣄⣤⣯⣀⡄⠀⢹⠀⣿⠈⣇⢸⠃⡟⠀⠀⡇⠀⠀⢾⠃⠈⠀⠀⠀⡀⢿⠋⠀⢀⡏⢳⣠⡏⠈⣿⡶⡄
⠀⠀⠀⠸⡇⠀⠛⠀⠀⡀⠀⠀⡇⢀⡿⠛⡿⠻⣟⠳⡄⠹⣦⠤⢸⡆⣷⠀⡏⣾⡼⠁⣀⡼⠃⢀⡄⢸⡄⠀⠈⠁⠀⠁⣿⠀⣤⣜⡇⠀⠹⡆⠀⣿⠀⣷
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠁⠀⢸⡇⢹⡇⠀⡇⠀⢹⢰⡇⢠⣧⠀⠘⡇⢹⢸⣧⡟⠛⠉⠁⢨⠟⢋⠙⢓⢷⡈⠀⠀⠆⢠⡇⠀⣧⠈⠳⡄⠀⢻⠀⡟⢠⡟⠋⡷
⠀⠀⠀⠀⠈⢧⠈⠀⠀⢠⠀⣼⠀⠸⡇⠀⢻⠀⡿⡼⢀⡞⠀⠀⠀⣇⣿⢸⣿⠀⠀⠀⠀⢻⣄⠈⠤⠈⠀⣳⣄⠀⠀⣼⠀⠀⠘⢧⡀⢹⠀⣾⢸⢃⡾⣠⠞⠁
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠃⠀⠀⠃⠀⠈⠓⠛⠃⠘⠀⠀⠀⠀⠹⠘⠟⠃⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉⠙⠂⠐⠃⠀⠀⠀⠀⠙⠊⠀⠃⠛⠚⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⠖⣟⣋⣛⣛⣛⣙⣋⣛⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠛⠋⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⣠⠞⠁⢀⣤⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠒⠉⠉⢦⡀
⠀⠀⢠⠞⠁⢀⡴⢋⡎⠀⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠛⠒⠒⠊⠀⠾⠤⠤⠤⢤⢤⡼⠦⠟⠸⠤⠤⠤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⣺⠖⠲⢤⡀⠀⠀⣓⠦⡀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⡇⠀⠀⠀⠙⡶⠋⠉⠑⡄⠀⠀⠀⠙⠲⢤⣄⡀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⡰⡁⠀⠀⠀⡇⠀⠀⠀⠀⢠⡤⠞⠁⠀⠀⠀⠀⡄⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⠢⢄⡨⠖⢷⣦⠤⠜⠁⠀⠀⠀⠀⠀⠡⠀⠀⠀⢠⠀⢸⢻⡤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⣞⠑⠒⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠤⠟⣒⣮⠔⡒⠤⣀⢊⡧⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠉⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⢳⠉⢀⡴⠋⢠⠔⡢⠠⠘⣉⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠶⡙⠞⠀⠈⢣⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢻⣛⢥⢫⠊⠀⠀⠀⢰⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⡤⢎⠄⠀⠤⠃⠀⠀⢸⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡖⠘⠒⠒⠒⠒⠒⢒⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣠⠋⡽⢿⣆⣠⡄⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢽⣿⣤⣄⣆⠀⠉⠈⠉⢟⢻⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠁⠀⠀⢠⣄⣴⣎⣁⡙⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠛⠷⠶⠝⠛⠛⠉

⠀⠀⠀⠀⢿⣿⣿⣿⠇⢸⣿⡇⢸⣿⡇⠀⠀⣿⣿⣿⠀⠀⢸⣿⣧⠀⣿⣿⠀⣿⣿⠀⢠⣿⡟⠀⣿⣿⠿⠀⢻⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⠀⢠⣿⣿⣿⡆⠀⢸⣿⣿⡀⣾⣿⠀⣿⣿⢀⣾⡟⠀⠀⣿⣿⠀⠀⣸⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⠀⢸⣿⠈⣿⡇⠀⢸⣿⣿⣇⢿⣿⠀⣿⣿⣾⡟⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣷⣾⣿⡇⠀⢸⣿⠀⣿⡇⠀⢸⣿⣿⣿⣾⣿⠀⣿⣿⡟⠀⠀⠀⠀⣿⣿⣶⠀⣽⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡏⢹⣿⡇⠀⣾⡟⠀⣿⣿⠀⢸⣿⡟⣿⣿⣿⠀⣿⣿⣷⡀⠀⠀⠀⣿⣿⠉⠀⣿⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⠀⣿⣷⣦⣿⣿⠀⢸⣿⡇⢿⣿⣿⠀⣿⣿⢻⣷⡀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⢸⣿⡇⠀⣀⣀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⢠⣿⡏⠉⢹⣿⡆⢸⣿⡇⣾⣿⣿⠀⣿⣿⠈⢿⣷⡀⠀⣿⣿⠀⠀⢹⣿⡀⣸⣿⡇⢸⣿⣧⣾⡟⠙⡇
⠀⠀⠀⠀⠀⠸⠿⠇⠀⠸⠿⠇⠸⠿⠇⠸⠿⠁⠀⠸⠿⢧⡼⠿⢷⣿⡿⠿⠀⠿⠿⠀⠈⠿⠷⠀⢿⣿⡆⠀⠈⠻⠿⠿⠋⠀⠸⢿⣿⡿⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⡟⠛⠛⠇⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣠⠏
⠀⢀⣠⣤⣤⣀⡀⠀⠀⣴⡿⠀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣀⣄⡀⠀⢀⣿⡿⠀⠀⣀⣀⠀⠀⣀⠀⠀⣸⣿⡷⠋
⣴⣿⠏⠀⠀⣿⣿⠦⠀⣹⠷⢻⣿⡿⠀⠀⣴⣿⠋⢨⣿⣿⠀⠀⣿⣿⠀⠀⠀⢀⣾⣿⠉⣹⡄⢸⣿⡇⠀⢠⣿⡿⠀⣼⣿⡇⠀⢹⣿⡇
⣿⣿⠀⠀⣸⣿⣿⠀⢠⡟⠀⢸⣿⡇⠀⣼⣿⡇⢀⣾⣿⣿⠀⢠⣿⣿⠀⠀⠀⣸⣿⡏⠉⠉⢠⣾⣿⣅⣤⢾⣿⡇⢠⣿⣿⡇⠀⣸⣿⡇⠀⠀⣼
⠘⠿⣦⠾⠋⣿⣿⣠⡞⠀⠀⢸⣿⣦⡾⠻⣿⣧⠞⠁⢿⣿⣴⠏⠹⣿⣦⣤⡶⠛⣿⣧⣠⡴⠋⣿⣿⡅⠀⠸⣿⣷⠟⠸⣿⣧⣴⠏⢻⣷⣤⡾⠁
⠀⠀⢀⣠⢾⣿⡟⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠁⠀⠀⣿⣿⢻⡀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠈⠁
⠀⣰⠟⠁⣸⣿⣇⣀⣀⣀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⣿⣿⣈⣧⣀⠀⢀⣀⣀⣀⡀
⢸⡇⠀⢀⣿⡟⣿⣿⡿⣿⣿⡄⠀⢸⣿⡇⠀⠀⣿⣿⣿⠅⢀⣾⣿⠿⣿⣦⠀⠀⣴⣿⡿⣿⣷⣿⣿⣿⣿⡿⠀⢸⣿⡿⣿⣿⣷⡀
⠘⣇⣀⣼⠟⠀⣿⣿⠀⠈⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⢸⣿⡇⢰⣿⡏⠀⢸⣿⡟⢿⣿⡿⠁⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⠘⣿⣧⠀⠸⠿⠇⠈⣿⣷⠀⠸⠿⠇⢠⣿⣧⠀⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣤⣼⣿⠇⠀⢸⣿⡇⠀⠀⣿⣿⣤⡀⠀⠹⣿⣷⡀⠀⠀⠀⠙⣿⣷⡀⠀⠀⢸⣿⣿⣤⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠛⠻⣿⣦⠀⢸⣿⡇⠀⠀⣿⣿⠛⠀⠀⠀⠙⣿⣷⡄⠀⠀⠀⠘⢿⣿⡄⠀⢸⣿⣿⠛⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⢀⣀⠀⠈⢿⣿⠀⠀⣀⡀⠈⢿⣿⡄⢸⣿⣿⠀⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⠀⢸⣿⡇⢸⣿⣿⠀⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣄⣤⣿⣿⠀⢸⣿⣇⡄⠀⣿⣿⣤⠀⠘⣿⣧⣄⣼⣿⠃⠈⣿⣷⣄⣼⣿⠇⢸⣿⣯⣀⠀⢸⣿⣇⣠⣾⣿⠇
⣤⣤⣠⣤⣤⣤⣿⣿⣿⣿⣿⣡⣤⣼⣿⣿⣧⣠⣿⣿⣿⣄⣤⣌⣿⣿⣿⣡⣤⣤⣌⣿⣿⣿⣩⣤⣸⣿⣿⣿⣤⣼⣿⣿⣿⣿⣫⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣩⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡄⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠚⠋⠉⠉⠉⠈⢆⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠶⠶⢶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢔⠛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⣤⣤⣌⣉⣉⡁
⠀⢀⣀⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠃
⠀⠀⠈⠉⠛⠳⢶⣒⡳⠾⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣠⠿⢿⣝⡢⡀⠀⠀⠀⠰⡄⠀⠀⡆⠀⠀⡴
⠀⠀⠀⠀⢀⣴⣶⣋⣽⢹⣖⠀⠶⢏⣁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣴⠓⠂⢰⣿⣷⣷⠀⠀⠀⣀⠃⠰⡄⣁⢲⠘⠁⠀⢀
⠀⠀⠀⠀⣿⠟⣡⠌⠀⠧⠤⣍⣓⡢⢌⣉⡒⠢⢤⣀⡀⠀⣀⣳⣀⣾⣤⠤⣀⣠⣿⣃⠀⠀⣀⡭⢤⣧⡀⣥⢀⣶⣦⡾⠁
⠀⠀⢸⣿⠽⠓⠲⠤⣄⠀⠀⠀⠀⢉⡟⠉⣹⣿⠋⠉⠉⠉⠉⠁⠀⠈⡿⠸⠶⣦⣭⢛⣢⣴⢳⣿⠦⠄⠉⢻⣿⣿⠏
⠀⠀⠸⣿⠀⠀⠀⠀⠈⠑⢽⡳⢶⡾⢠⢠⣿⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣏⣿⡯⠍⠀⠀⢠⠛⠁
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠈⠑⢧⣸⣼⣿⣿⣦⡙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣼⣟⠍⠀⠀⢠⠃
⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣛⠿⣯⣛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢳⣿⠂⠀⠀⢠⣧
⠀⡇⠀⠀⠈⢧⡀⠀⠀⠀⣀⡠⠤⠒⠋⠉⠀⠀⠀⠉⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⠇⠀⠀⢠⣿⡿
⠀⣧⠀⣇⠀⢰⣷⠔⠒⠉⠁⠀⠀⠀⢀⣀⣤⠤⠤⣤⡤⢤⣿⠛⠀⠀⠀⠀⠀⢰⣿⣾⡇⠀⠀⢠⠿⠛⠁
⠀⣿⠀⣿⠀⠀⠁⠀⠀⠀⠀⣠⢴⡚⢿⣿⣿⠂⠀⣿⣿⠄⢿⡖⡄⣦⠱⣄⠀⢸⣷⠟⠀⠀⢠⠋
⠀⣿⡀⢸⡄⢸⡀⠀⠀⣠⣾⡻⣦⣻⣎⣿⠏⠀⠀⠘⠋⠰⣿⣷⣿⣼⣧⡹⣦⡸⣿⡷⠶⣶⣿
⠀⣿⡇⢸⡇⠘⣇⠀⣴⣿⣌⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢹⣶⣭⣙⣛⣛⣛⣒⣛⣛⣻⣽⣆
⠀⣿⡇⢸⣿⠀⣿⣴⠛⠿⣿⣿⣿⠟⠉⠁⠀⢀⠀⣄⢰⡀⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⣆
⠀⣿⣿⣸⣿⣄⣿⣇⣀⣀⣀⣈⣟⣀⣀⣀⣀⣸⣇⣿⣈⣇⣸⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣻⣸⣿⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠇⠀⠀⠀⠀⠀⣠⣤⠶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢀⣤⣤⡾⠉⠀⠀⠀⠀⠀⠉⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢰⡏⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣠⡼⢷⣄⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠉⠀⠀⠙⣦⡀⠹⣦⠀⠀⠀⠀⠀⠀⢀⣴⠶⠾⠃⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠖⠀⠀⠈⢷⡄⠈⠻⠶⣤⣤⣤⡶⠟⠁⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⢀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠉⠙⢷⣄⢀⣤⢾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⢻⡟⠛⠛⠛⠉⠀⠀⠀⠀⣀⣴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠙⣿⣅⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⣀⡀⢀⣤⠞⠋⠙⢷⣄⠀⠹⣦⡀⠀⠀⠀⠀⠀⢀⣤⣤⡴⠶⠾⠿⠖⠛⠛⠛⠉
⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⣀⣴⠞⠋⢻⣟⠁⠀⠀⠀⠀⠙⢷⣄⠈⠻⣦⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⣙⣷⡀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣳⡶⠛⠁⠀⠀⠀⠀⠀⠀⣴⠞⠛⠳⣦⡀
⠀⠀⠀⠀⠈⠻⠶⠶⠾⢻⡿⠀⠀⠀⢠⡾⠛⠉⠉⠻⣦⡀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⠛⠷⠶⠶⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⣾⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣄
⠀⠀⠀⠀⢰⡄⠀⢸⡇⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠘⣧⠀⠀⠙⠶⠴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠷⢦⣤⣤⣀⣀⣀⣀⣤⣤⠶⠖⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋


⡴⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦
⠧⣴⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹
⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⡀⠀⡀⣀⣀⡀⢀⡀⠀⣀⣀⠀⠀⠀⠀⣀⣀⠀⠀⢀⡀⢀⣀⣀⢀⣀⡀⠀⢀⣀⠀⢀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠤⠀⡇⠀⠹⡀⡇⠈⡇⠀⡜⣣⠀⣯⢬⡃⡇⠀⠀⡧⠄⠀⠀⠧⣀⠀⢹⠀⢸⣅⡝⢰⠁⠀⢳⢸⣴⡁⢸⡧⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠛⠉⠉⢳⡄⠀⠀⠀⠀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠶⠦⣤⣀⣀⣀⡇⠀⠀⢀⣤⣾⠙⢲⣮⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⣠⠞⠛⠲⢤⡈⠣⠉⠈⠷⠀⠀⢸⠃⠀⢀⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠙⢦⡀⠀⠀⠈⢁⠶⠤⠤⠤⠤⡟⠳⣄⣹⠏⠀⢈⠽⠓⢦⡀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠦⣿⣤⣤⣤⣯⡀⠀⠀⠀⡲⣬⠶⠆⠁⠀⠰⣧⠠⢤⣬⣭⣽⣶⣦⣤⣀⣀⠀⣹⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⠁⠉⠒⠢⠶⠛⠁⠀⠀⠀⠀⠀⠈⢆⣀⠀⠱⡀⠀⠀⠀⢩⡟⠋⢹⣟⡋⠻⠚⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠹⡟⠛⠓⠶⠤⣼⡀⢸⠈⠩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢹⡀⡇⠀⠀⠀⠀⠀⠙⣾⣅⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠉⠙⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣈⣷⡇⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡻⠇⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠴⠶⠶⠶⠶⠶⠶⠶⢦⣤⣤⣀⣈⣷⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠖⠚⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠙⠻⠧






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠞⠁⠯⠅⠪⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠆⣂⠡⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⣒⠲⠦⢚⣫⣭⠔⠚⢃⣁⠤⠄⠀⠤⠄⡀⠀⠀⠀⠀⠀⠠⠖⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠤⢒⡩⠔⠂⢉⡀⠤⠤⠘⣛⣃⣀⡈⠉⠐⠒⠲⣜⠃⢰⣒⠻⣶⠂⠉⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡡⠚⡡⣒⠥⠐⢂⡩⠤⢴⣲⡚⣏⠙⣌⣱⠬⠟⠒⠀⠀⠉⢳⡄⡈⠑⠦⡈⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡞⡡⠞⠉⢀⡀⣨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⡀⠈⠢⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠆⠀⡴⠋⢀⡤⠊⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡚⢆⠀⠈⠢⡀⠘⠄
⠀⠀⠀⠀⠲⠄⢠⣄⠎⢀⡜⠀⢠⢊⡠⠂⠁⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣍⢣⠀⠐⢌⢄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⠀⡰⠋⠁⠀⠀⠀⠀⠘⠤⣀⣀⡧⠴⠃⢀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡂⠷⠀⠈⡆⠱⣰
⠀⠀⠀⠀⠀⠀⡀⣇⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣃⣈⠦⠯⣆⡝⡧⢋⡺⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣉⡇⠀⢓⠂⢻⠢⡀
⠀⠀⠀⢀⣴⠀⠀⠁⠀⡌⢰⠆⡰⠆⠀⠀⠀⡠⠖⠊⠉⢸⠀⠈⠵⠀⠀⡇⣃⡌⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡁⠀⢀⡄⠹⢆⠈⠄
⠀⠀⠀⡎⢸⠀⠀⠀⠘⠀⣎⢎⠴⠀⠀⠀⠘⠤⠤⠂⢀⢴⢢⠀⠀⠀⠐⠁⠁⠣⢔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠸⢣⠀⠄⠉⠺
⠀⠀⠀⡁⡈⣀⢰⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢸⢼⠖⡆⢀⡀⢀⢀⡀⣿⡄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⡄⠰
⠀⠀⠀⢇⡁⡏⠘⠲⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⢂⠸⠀⠷⠣⠓⠊⠋⣇⡟⠳⠋⢃⢉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡅⠀⠆
⠀⠀⠀⠘⠃⡇⠰⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠙⠀⠀⠀⠀⠀⠀⠀⠁⠐⠤⠊⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠋⠒
⠀⠀⠀⠀⠀⢩⠄⢡⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢖⡂⢠⠂⡄⢰⠴⣀⢷⢚⠀⠂⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠄⣼
⠀⠀⠀⠀⠀⠈⠈⠄⠳⢥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢉⣋⣉⡁⠀⠀⠀⠸⡠⠋⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⠜⠶⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠁⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠉⠳⡄⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠡⢟⡍⠀⠀⠤
⠀⠀⠀⠀⠀⢀⢀⣘⡌⠿⠆⠐⠧⠐⠷⢌⡁⠢⡀⠀⠀⠀⠀⠙⠒⠤⠤⠤⠼⢦⠒⠉⣂⠁⣦⠀⠀⠀⠀⠀⠀⠀⡈⠀⡰⡡⠡⡘⠁⢀
⠀⠀⠀⠀⠀⠈⢁⡸⠿⠀⠀⠘⠴⠀⠲⣆⠈⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⣰⢡⠾⠊⠠⠣⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠈⡴⢠⠀⣤⠤⢀⡑⠢⢄⡹⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⢪⠃⠀⢀⠔⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣈⡑⠂⢤⣈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡡⢊⣀⠔⣹⣤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⢀⡀⠉⠒⠤⠄⣀⠀⠀⣀⣀⠙⠟⠴⠚⠯⡒⠕⠈⣡⠟⠨⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⣥⢛⡱⢦⠄⣀⢀⣄⣭⡌⠩⠞⠋⡡⠊⠀⡠⠚⡁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠱⠖⠉⠑⠶⠄⠀⠤⠄⠘⠒⠈⠁⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠴⠶⠶⠶⠶⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣭⠀⣴⣿⡄⢸⣿⣦⣿⢸⣧⡾⠃⢸⡖⠶⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⢸⡧⢶⡿⢰⡿⠺⡷⠸⠇⠹⠿⠸⠏⠻⡄⢸⡏⠛⢸⣧⠈⣷⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠇⣈⣁⣬⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡙⠲⠿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣤⡄⣨⣭⣭⡉⠉⢩⣭⢩⣭⣭⣭⢩⣭⣭⡍⣭⣭⡍⣭⡍⢠⡄⣤⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣬⣅⣿⣧⣼⠇⠀⣿⢻⡇⢸⣿⠀⢸⣧⣤⠀⣿⣤⡄⣿⡇⢸⡇⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⡇⢼⡏⢿⡄⢸⡟⠛⣿⠸⣿⠀⢸⣧⣤⡄⣿⠀⠀⢿⣧⣼⡇⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣦⣴⠶⣦⢰⡆⠀⣶⠶⠆⣴⠶⡆⣴⠶⠦⢰⡶⠆⣶⠶⣦⢰⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⠓⣿⢸⡇⠀⣿⠓⠂⣛⢲⣆⢙⠓⣶⢺⡗⠂⣿⠀⣿⢸⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣋⣙⣛⣋⣘⣛⣃⣙⣛⣃⣙⣛⣁⣉⣛⣋⣘⣛⣃⣙⣛⣋⣘⡛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠉⠙⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣀⣤⡀
⠀⠀⠀⠀⣠⠊⠀⡠⢶⠋⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⢦
⠀⠀⠀⣾⣁⣀⡞⢠⡏⠀⠀⠀⠀⣠⣀⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡽⢹⣭⣍⡀⠈⠉⠹⣍⠉⠉⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡏⠀⠀⠉⢢⡴⠚⠓⡍⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢀⡠⠏⠁⠀⠀⠀⣄⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠘⠦⢀⠰⠞⢶⡄⠴⠃⠀⠀⠀⠀⠀⢂⠀⠀⢀⡄⢀⡿⣄⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢞⠁⠒⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡤⠜⣓⠬⡔⠒⠢⣠⢩⠧⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡈⡏⢀⡔⠁⢠⣊⠕⠀⠭⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣀⣀⣀⣀⣀⠤⣤⣴⠒⠛⠃⡝⠁⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠯⠓⠱⠁⡄⠀⠀⡇⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠵⠋⠀⠉⠀⠀⠀⡇⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢧⣀⣀⡀⣀⡤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⠞⣻⣦⡀⣀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⣯⣍⣘⡄⠈⠋⠛⢶⢶⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⣠⢤⣤⣃⠑⠢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⣤⢦⢀⣿⡇⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⣿⡡⠊⣸⣿⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⣿⡏⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢠⣿⡧⠔⣉⡀⠀⠀⢀⣤⠤⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠞⠀⣿⡆⢰⣿⢁⠔⠀⢼⡏⡿⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⢹⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⢀⣿⡗⣿⣿⠁⠀⠀⠈⡳⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣿⣧⠔⢩⡿⠁⠘⢿⡄⠀⡠⠚⢀⣤⠖⠒⡄⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠁⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⡇⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⡠⠃
⢀⡀⠀⢠⡶⢦⠀⢸⣿⠁⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⢀⠎⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⢀⠔⠁
⡞⠀⠀⣿⠃⠎⠀⢸⣿⣀⠎⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⢀⡾⠃⠀⠀⠀⢀⣾⡟⠀⠀⢀⠄⠁
⠳⡀⠀⠹⠮⠀⠀⠸⣿⡅⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠸⣇⠀⠀⠀⠀⣸⣿⠁⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠰⠇⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⢀⣤⠤⡀⣸⣿⠀⠀⠀⠀⠀⠈⠑⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⣠⡖⢦⢀⣴⠆⠀⢀⣿⡇⠀⠀⣠⠤⣄⠀⠀⠀⢸⣿⠀⠀⣰⣿⠁⢀⠃⣿⡏⠀⠀⢠⣶⠀⠀⣴⡖⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⢰⣿⠃⠈⣼⣿⠀⠀⢸⣿⠁⢀⣾⡏⠀⣽⣿⠃⠀⢸⣿⠀⢠⣿⡇⠐⠀⢀⣿⡇⠀⢠⣿⡇⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢰
⠀⢸⣿⡀⠀⣿⣿⠀⠀⢸⣿⠀⢸⣿⠀⠀⣼⣿⠀⠀⣼⣿⠀⢸⣿⠁⠀⠀⢸⣿⠁⠀⣸⣿⠀⢠⢿⡟⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⡜
⠀⠀⢻⡅⢠⢿⣿⠀⠀⠸⣿⠀⢸⣿⠀⢠⢻⣿⠀⢰⠃⢿⡄⡞⢿⡄⠀⢀⢻⣿⠀⡴⠉⠿⠴⠃⢸⡇⠀⡜⠈⢿⡄⠀⠀⠀⠀⡰⠁
⠀⠀⠈⠛⠊⠈⣿⡇⠀⠀⠙⠃⠊⠈⠓⠃⠈⢿⡀⡾⠀⠈⠛⠁⠈⠳⠠⠎⠘⣿⡎⠀⠀⠀⠀⠀⠀⠙⠚⠀⠀⠈⠳⣄⠀⡀⠜⠁⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⠠⠀⠂⠐⢿⡇⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⣠⢤⣄⠀⠀⣻⣧⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠃⠠⠐⠁⠀⠸⣧⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁
⠀⣸⠁⠀⣹⣿⠁⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠋
⠀⣿⠀⢸⠀⢿⡇⠀⢸⣷
⠀⠸⡀⠘⠦⠾⠁⠀⠀⣿
⠀⠀⠱⡀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠈⠢⠀⠀⠀⠰⠋
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢤⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢈⠻⠋⣨⣀⠤⣤⣀⣤⣀⣻⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢨⣿⣿⣿⣿⣷⣦⡀⠀⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⠀⢨⣿⣧⠈⢻⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⠁
⠀⠀⠀⠀⠀⠀⠀⢠⢉⠀⠠⠘⣇⣿⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⡛⢻⠟⢛⠛⠉⣹⡿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⠀⠀⢇⣿⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢢⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠁⠈⠀⠀⠀⠈⠉⠀⠻⣇⠀⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡿⢿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⢸⣿⣿⡗⣷⡄⠃⠀⠀⠀⠀⣀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⢱⣆⣠⣤⣶⣟⣀⣿⠇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠘⠏⠸⠗⡎⢠⡅⠋⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣶⣦⠙⠙⠁⠀⠉⣥⣴⡿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣼⠀⠀⠈⠁⠒⠓⠠⠟⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⣿⣿⠍⡹⠀⠸⠀⠀⠀⠚⠁⠾⠿⠇⡀
⠀⠀⠀⠀⠀⡄⣸⡍⣖⠤⠀⣀⣀⡤⠓⠐⣾⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣃⣃⠀⠂⠀⢣⣤⣐⠀⠀⣸⣄⣴⣶⣶⡎⢁
⠀⠀⠀⠀⠀⠀⣿⣸⠇⠀⠺⣿⠹⡇⠀⠀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣻⡿⠟⢋⠁⠀⠀⠀⣸⠟⡘⠀⠀⠁⢩⣿⣿⣿⠀⠈
⠀⠀⠀⠀⠛⢈⣿⡟⠀⠀⣾⣿⠀⣷⠀⠀⢴⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⢹⡇⠀⠈⠙⠒⡶⠀⠉⠄⢀⣌⣀⣠⣾⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠠⢸⣿⡏⠀⣿⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⣀⣀⠈⣿⡇⠐⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⣞⢸⡇⢰⠠⣾⡇⠀⣿⠀⠀⠀⢿⡿⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡍⢿⣿⣿⣿⡃⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠃
⠠⠀⠀⠀⠀⡟⣼⠃⡟⠀⢿⣇⠀⠸⡇⠀⡀⢸⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡈⣿⣿⡟⠁⠔⠀⢸⣿⣿⢿⡟⣾⣿⣿⣿⣿⡄
⠐⠃⠀⠂⠀⣿⣿⣠⣿⢀⣤⡿⠀⢀⣧⣀⣁⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠿⠓⠉⠉⠀⠀⠀⠀⢰⣿⢇⣾⣧⣿⣿⣿⣿⣿⣟
⠀⠀⠈⠀⠀⠛⠳⠿⠿⠛⠿⠷⡶⠿⠟⠛⣛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⢰⣶⡆⠀⠀⢀⣾⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠠⠀⠀⠀⠔⠀⠛⢀⠋⢹⣿⣿⢃⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡠⠈⠁⠀⡁⠀⢉⡀⠀⠀⠀⠐⠒⠒⠉⠀⠀⠐⠒⠒⠰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⠠⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⣽⣿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠙⠿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠈⠉⠛⠿⣿⡿⢿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠤⡆⣦⣄⡠⣤⣤⡄⣤⡤⢰⠠⣄⣤⠤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠈⠁⠁⠀⠀⠀⠀⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠄⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣇⣇⣿⣸⡗⣿⣁⣀⣀⣤⠖⠃⠀⠀⡀⠀⠰⠋⠀⣀⣠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠻⢾⡟⠙⠉⠛⢁⠈⠙⠋⠉⠀⠀⠀⠀⣾⢙⠀⠀⠊⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⠀⢀⣾⡇⣠⣿⠄⡇⠀⠀⢠⢦⡀⢠⡶⡀⣿⣮⡀⢺⣇⠀⠉
⠀⠀⠀⠀⠀⠀⠒⠚⠛⠉⠉⢁⡀⠀⢸⣼⠇⠀⣿⠀⣿⢴⡄⣿⣸⣇⣸⡇⢇⢽⡿⠤⢺⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠻⠴⠹⠆⠳⠋⠋⠈⢁⡁⠀⠀⠀⠀⠛⠃⠀⢰⠋⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⠈⠉⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢦⡁⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠉⠒⠤⠤⣤⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡠⠤⠒⠂⠙⠖⢊⠉⠀⠀⠐⠤⡀⠈⠙⠲⣄
⠀⠀⠀⢤⡀⠠⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⢠⠎⠀⠀⠀⠑⢦⡀⠀⠀⠈⢶⡀⠀⠈⢳⡀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⡸⢗⠋⢉⢢⠀⣠⠊⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣿⠀⠀⠀⣷⠀⠀⠀⡈⠾⠥⠤⢤⡐
⠀⡔⢰⡃⠈⠆⠀⡧⢠⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠏⠀⠀⠀⡿⠀⡰⢫⠀⢦⡀⠈⢢⠹⣆
⠀⠁⠈⠃⠀⠀⠊⠁⢸⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡾⠁⠀⡇⣿⠀⢸⡇⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡴⠋⠀⠀⠀⢳⡘⢇⣾⠇⣀⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⢀⣀⠴⠒⠦⢤⣀⣀⣠⣴⣯⣤⡴⠞⠋⠀⠀⣀⠤⠐⠒⠛⠂⠁⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡨⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⣠⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⡀⠀⢻⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣆⠸⣿⣿⣿⣿⠿⠆⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣤⣾⣿⣿⣷⡉⠛⠛⠉⠀⢀⣠⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡦⠰⣦⡙⣿⣿⣿⣿⣿⣿⣿⠾⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⣛⣍⢩⣾⣿⣮⡻⣮⡻⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⢡⣾⣿⠛⠃⠙⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣰⡿⢟⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⣿⡇⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⣿⡇⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠠⢦⣤⣤⣤⣤⣤⠦⠀⠀⢀⣾⣿⣿⣿⣷⣭⣵⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⣀⣀⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡏⣶⡎⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⠶⠾⠿⠿⠿⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢉⡀⠤⠉⢠⠄⠂⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠆⠀⠒⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠀⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡔⠂⢲⢲⠦⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⡀⠀⠐⠊⡂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠛⡅⠀⠀⠀⠁⢂⣇⣟⠀⣀⡀⣀⡀⣾⢁⡟⡀⡀⡾⠃⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠂⠚⠘⣹⣚⡛⠚⠙⣛⡓⢺⠉⢛⣛⣃⠀⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⡀⠚⠀⠀⠀⠀⠀⠀⡴⢄⡦⡄⣤⠐⡶⢠⠖⣴⠂⡄⡄⡄⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⡧⡄⣷⠁⡿⡇⡇⢸⠆⣿⠘⡇⡇⡇⠀⠀⠀⠀⠸⢀⡀
⠀⠀⠀⠀⠀⠐⠂⡁⡀⠀⠀⠀⠀⠀⠙⢃⡋⠛⢃⡃⠃⠘⠒⠉⠀⠙⡡⠗⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠩⡅⠀⠀⠀⠀⠀⢀⡥⡇⡰⡡⠃⠂⠀⠐⠠⠄⡴⠤⠃⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚⠀⠀⠀⠀⡼⠄⡴⠏⣰⡢⣮⢴⡵⡣⠶⡂⡀⠀⠀⠀⠀⠐⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢁⠀⠀⠀⠁⠈⠀⠉⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠠⢄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣕⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢃⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠆⠀⡀⢀⡀⠐⠆⠀⡀⠒⠂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠠⠄⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠿⠿⣿⣿⣿⠿⢿⣿⠟⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢹⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⣀⣀⣀⣾⢻⣇⣀⣠⣴⣶⣦⣀⣴⣷⣄⣀⣈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣷⣦⣴⣏⣠⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⢾⣷⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠈⠋⠀⢻⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣹⣧⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣤⡀⠄⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣈⣟⣻⣷⣾⣶⡞⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⣽⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣏⣹⠛⠉⣽⣿⣿⣿⢿⣿⣾⣿⣿⣿⣿⣏⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡀⢘⣿⣿⣿⣿⠟⠛⢿⠿⣿⣿⣿⠿⣶⣤⣨⠽⣿⣿⣿⣿⠉⠛⠋⠉⠁⠙⠛⠁⠀⠀⠀⠀⠀⠉⠉⠉⣛⣻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿⣿⣿⣿⣀⣤⣠⣤⠀⠀⠀⠀⠘⢿⡟⠿⠿⢿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⠙⠛⠛⡛
⠿⢟⠉⠀⠀⠀⠛⠛⠛⠛⢉⣿⡿⠋⠻⣿⣿⣿⠏⠉⠁⢩⣻⣷⠀⠀⠀⠀⠀⠙⢿⣴⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⢿
⠀⠌⠀⠀⠀⠀⠀⠀⢠⣶⠟⠁⠀⠀⠀⡜⠛⠁⠀⠀⠀⣿⣿⣿⣿⣦⣤⡀⠀⠀⣬⣍⣱⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠀⠀⢰⡇⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡇⠀⢀⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠘⠛⠙⠛⠀⠀⠀⠀⠸⠇⠠⠀⠀⢁⣼⣿⢻⣿⣿⠟⠉⠀⠠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⠀⣼⣿⡟⠈⠋⠁⠀⠀⢀⣴⡏⢻⣿⣿⣿⣧⣿
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣾⡗⢸⣿⣿⡁⠀⠀⠀⠀⠀⣘⠁⢻⣿⣿⣿⡃⣿⡏
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢿⣿⣿⠁⣸⣿⠿⠿⠿⠿⠿⠶⠿⠿⣿⣾⣿⣿⣿⢧⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠠⠀⠀⠀⠀⠀⠀⠀⣴⣤⡀⠛⠻⣿⡿⠉⠠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⢈⣋⣻⣿⠃⠀⠸⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⠛⢁⣀
⠀⠖⠀⠀⠀⠀⠀⢰⣿⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⠈⠉⠀⠰⠶⣶⣦⠾⠷⠶⠶⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿⠿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠻⢿⡿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠉⠓⣾⡿⣤⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⣀⣀⣾⣿⣿⡿⠛⠁
⠀⠠⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠘⢷⣞⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⣀⣠⣾⣿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠙⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣾⣿⣿⠟⠁⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣴⣿⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠓⠈⠛⠉⠀⣀⣴⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⢰⣿⣹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣶
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⣸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿
⠀⠀⠀⠀⢀⣠⣿⣿⣿⠇⠀⠀⢠⣿⣧⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⠿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣿⣿⣷⣦⣀⣀⠚⠛⢛⣿
⠀⠀⢠⣶⣿⣿⣿⣿⣿⣀⣠⣤⣼⣏⣽⣇⣀⣠⣤⣀⣠⣤⣰⣿⣿⣯⣭⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣠⣽⣿⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠹⣿⣿⠿⠿⢿⠿⠟⠉⢠⣽⣿⣏⣿⠿⣿⣿⣿⣿⣿⠛⠛⠋⠁⠀⠀⠀⣤⠀⣴⡄⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⣤⠀⡀⠀⠀⠀⠀⢈⡟⠉⠁⣬⣟⢩⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠁⠀⠀⠀⢀⣄⣹⣿⣿⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣶⡶⠿⠿⣿⣇⣿⠁⣠⣿⣿⣿⣿⡿⢿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢿⡈⠉⠁⠀⠀⠉⠉⠀⠰⠉⢹⣿⣿⠿⠛⠂⢉⣀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⣴⣦⠀⣶⣶⣷⣿⠟⠁⣴⣾⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣬⣭⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣀⣈⣉⣠⣀⣀⣀⣈⣀⣠⣤⣬⣥⣴⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣟⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢄⣑⢤⡢⡀⠀⠀⠀⠀⠀⡴⠋⠉⠑⣄⠀⠀⠀⠀⠀⠀⢀⠀⡘⠃
⠀⠀⠀⠀⠀⠬⡪⢲⠀⠀⠀⣸⠁⠀⠀⠀⠘⡄⠀⠀⠀⠀⡠⡫⡪⠂
⢀⡞⠁⠙⡀⠆⠈⠒⠀⢀⣠⣇⣀⡀⠀⢀⣀⣡⡀⠀⠆⢈⠌⠜⠀⠀⠀⠀⢀
⠈⠀⠋⡢⢐⠄⠀⣴⢊⣉⠀⠀⠀⠀⠀⠄⠀⠀⣀⣉⠒⣤⣆⠀⡐⠃⠀⠒⠉
⠀⠈⠑⠢⣱⠗⠊⠁⠀⠀⠀⠀⢀⡤⣀⡐⠤⠊⠹⠿⠀⢀⢄⡹⠁
⠀⠀⠀⠀⡏⠁⠂⡀⠧⠄⣀⣀⡼⠀⠀⢠⡤⠀⠆⠀⠒⠈⣽⠋⢣⠈⠋⠀⢀⡀
⠀⣠⠚⠀⠛⠄⠀⠁⠀⣜⠁⠎⠇⡀⣀⣀⣠⣤⣬⣤⣁⠀⡀⠋⠁
⠀⠉⠲⠀⠒⢣⠀⢀⠀⢠⠆⠀⢈⡴⠛⠋⢉⣽⣟⡙⢻⣾⡄⡈⠂⠐⠒⠀⠀⢤⣄⡀
⠀⠀⡀⠈⡁⠊⢇⣩⠑⠀⢁⠸⠃⠀⠀⠀⢈⣿⠟⠀⠈⠁⢻⢷⠀⠤⢢⠀⢄⠀⠑⡌⢢⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠈⡆⠉⠀⢀⠀⠀⢀⡼⣇⣀⠀⠤⠀⠄⣀⠀⠰⡀⠀⠂⠀⠀⡁⠀⠙⠲⠦⠶⠖⠒⠒⠲⢤⡀
⠀⠀⠀⠀⢀⢦⣠⠶⡔⠃⠀⢖⡁⠀⠐⣿⠏⢿⢿⡆⣸⠀⠀⠈⠠⠲⠃⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⡆
⠀⠠⡀⠐⠉⢀⠄⠀⢳⠀⠀⢸⡱⡿⣿⠫⣶⣶⡾⠃⠀⠀⠐⠠⣁⠀⠀⠀⠀⡀⠀⣠⣤⠄⢰⣆⢾⡙⢻⣟⡉⠀⠀⠻
⠀⢔⡡⠊⠔⠁⢀⣀⠈⢣⡀⠀⠋⠉⠲⢯⡎⠀⢀⠄⠀⠀⠠⠗⠀⠀⢴⡖⢦⣿⠀⢸⡷⠆⣿⢽⣤⣽⢾⣯⡁⣄⠀⢸
⠀⠉⠀⠀⠛⠊⠉⢸⡀⠀⠈⢿⡁⠀⠀⠀⠀⡽⠁⠀⠰⠂⠀⠸⠀⠀⠘⡷⠋⠘⠷⠂⠛⠃⠉⠀⠉⠀⠀⠀⠀⠁⠀⡤⠄
⠀⣤⣤⣤⣄⠀⠀⢀⣹⣦⣤⣤⢹⣥⠤⣤⢭⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠛⠀⠀⠀⠀⡀⢀⣀⢠⠄⡖⠀⡆⣾⡆⠀⡇⢀
⠀⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣧⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⠤⣰⠂⠀⡏⠂⣿⡜⡶⡘⣃⠸⠀⠣⠜⠓⠂⠂⢸
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣟⡾⣅⠀⠳⠆⠃⠈⠁⠁⠉⠀⠀⠀⠀⠀⠀⢀⣾
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢸⠇⠀⠂⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠶⣤⣄⣀⣤⣤⠴⠶⠾⠿⠿⠿⠿⠿⠿⠿⢿⠿⡿
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⣀⡴⠶⠒⠲⠶⣤⡶⠚⠛⠛⠛⠳⠦⣄⡀⠀⠀⠀⠀⠀⢀⡾⠋⢧⡀
⢀⡼⠋⡀⠀⠀⠀⢀⣀⡿⠀⠀⠙⠛⠳⣦⠈⠻⣆⠀⠀⠀⣰⠟⠀⠀⠈⠻⢦⡀
⣼⠁⢰⠏⠀⠀⣰⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢀⡾⠃⠀⠀⠀⠀⠀⠀⠙⢦⡀
⣿⠀⣾⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢿⡀⢸⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣠⡟⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠘⠛⠛⠛⠛⢳⡟⠀⠀⠀⠀⠀⠀⠺⣍⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⣀⣀⣠⡿⠀⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢀⣠⣤⠤⠴⠶⠶⢭⣭⣭⣿⣥⣤⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣭⣿⣶⣦⣄
⠀⢰⡏⠁⠀⠀⠀⣀⣠⡞⠉⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠴⠞⠋⠉
⠀⠀⠙⠶⣤⣀⠀⠉⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠶⢦⡴⠶⢿⡄
⠀⠀⠀⠀⠀⠉⠛⠓⢿⡀⠀⣿⠀⠀⠀⠘⠛⠲⣶⠶⠶⠦⢤⣤⣤⣤⣄⣀⣀⣀⣾⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢸⡆⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢻⡶⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣧⠀⠀⠀⠀⠀⣿⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⣦⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢳⣄⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠙⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⣄⠀⠀⠀⠀⠀⠀⠘⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣉⣛⣻⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣶⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⣀⡀
⠀⠀⠀⣀⣠⣤⣀⣀⣀⣀⡈⠿⠿⢿⣿⣟⠃⢹⡄⠀⠀⢀⠀⠀⠀⠤⠤⠄⠀⠀⢀⣀⣀⣀⣴⣷⣿⣿⡟⠋⠉
⢀⣴⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⠇⠀⠀⠀⠀⠀⣠⠴⣖⣻⣯⣭⣭⣽⣿⠿⠿⣿⣿⠿
⠈⠻⠿⢿⣿⣿⣿⣿⣟⢋⣭⣉⢉⣹⠛⢋⠁⠤⠶⣺⣿⣿⣯⣍⢁⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⢰⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠹⣿⣿⣿⣿⣿⢋⡼⠋⠀⣠⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡷⠋⠀⣀⣤⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠁⠀⢈⣡⣾⣿⡿⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠏⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠿⢿⡿⠉⡽⠁⠀⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⣿⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠐⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣠⠂⠀⠈⠀⠀⠀⠀⠁⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠐⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠉⠁⠀⠉⠀⠸⠀⠀⢠⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠲⠀⠐⠀⣀⡀⠈⡁⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⣍⠐⠀⠐⢾⣀⠀⠀⢀⠴⠆⣠⠀⣤⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⠀⠀⠉⢆⡐⠂⠀⠻⠇⠁⠀⠰⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢱⡀⠀⢀⡀⢀⣰⠎⣵⣤⠖⠂⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠦⣸⠃⢸⡥⠀⠈⠉⠘⢠⣄⠀⠀⠀⠀⢦⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡓⢤⠃⠀⢼⠃⠀⠀⠲⣀⢀⠀⠋⣤⠋⠀⠀⠻⢤⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠄⠀⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡞⠙⢿⡀⠀⠀⣠⡾⠄⠀⠈⠀⠘⢘⣯⠀⠀⠀⠒⠀⠀⠉⠀⢷⣄⠀⠀⠀⣠⣿⠟⠃⣀⣠⢶⣉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠖⣀⠑⠘⣷⠀⠛⠈⢣⣶⠇⠀⠀⠀⠀⢿⡟⠁⠀⠀⠀⠀⢲⡟⢦⡙⢟⠉⣿⣿⡿⠛⠛⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠿⠀⠸⢇⠀⠀⠀⠘⠻⠆⠀⠀⠀⠀⠼⢡⣦⠀⠀⠀⢀⣾⢃⣾⠧⠈⢷⣿⡿⠛⠂⢀⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⡈⠃⠰⣶⠀⠀⠀⠀⠰⠶⠒⠶⠄⠀⢀⠀⠠⠄⠰⣿⡛⠓⢀⣾⡧⠤⣿⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⡀⠀⠀⣱⣄⠀⠈⢄⡀⠀⠀⠀⠀⠀⠀⠈⣿⣋⣿⣡⢀⣴⣿⠿⣤⣋⣻⡷⠿⢧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣬⣿⢱⣄⠈⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠉⠻⠿⠐⠚⠏⠀⠴⣤⡙⡅⠘⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣶⠏⠣⣾⣗⣇⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡶⠄⢤⣀⣼⠃⠐⠋⢁⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡇⢸⢡⣿⣷⣾⣿⣾⣷⠀⠁⠀⠀⠀⢀⡤⠋⠀⠀⠀⢠⠿⢿⡇⢠⣶⢈⠀⠀⣀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣷⣼⣿⣇⠹⣇⢿⠻⣟⣄⠀⠀⠀⠀⠈⢡⡴⠃⣠⠛⠁⠘⢦⠨⠄⣣⠌⠀⢢⡴⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠰⠃⠠⣿⣿⡏⠸⣧⣿⣄⣠⣿⡽⡇⠀⠀⠀⠀⠀⠐⣴⣏⠀⠀⠀⠘⢿⠄⢁⠠⡌⠁
⠀⠀⠀⠀⠀⠀⢠⣐⣄⣰⣿⡏⢻⡀⢻⣽⣟⠛⢻⣷⣬⣦⣦⡠⠀⠀⠀⠻⠏⣳⣄⠠⡆⠀⠀⢛⢢⠘⠦
⠀⠀⠀⠀⠀⠀⠈⣽⠋⡉⢻⣷⠈⠁⠰⣿⢹⣷⣆⣀⠀⣈⠛⠓⣦⣀⡐⠈⢀⠻⣦⢣⡀⠀⠀⠀⡈⠂
⠀⠀⠀⠀⠀⢀⣰⣇⣈⢁⣼⠿⠆⢺⡀⠿⢶⡞⢯⠛⠿⣿⣾⣿⡿⠻⣿⣆⣼⠷⡌⣷⣤⣀⠀⠀⠀⣴⡿⠐⠋⠁⠐⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠚⠛⠉⠉⠁⠛⣤⣷⡀⠈⣷⣶⣄⢀⢸⣿⢂⣤⣀⣤⣴⠆⠀⢿⡙⣶⠻⠍⡏⠀⠳⠄⡀⠀⠑⣤⠘⠲⠀⠀⢀⡄⠀⠀⠀⠰⣇⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⣯⠻⣷⠛⠀⠀⠀⡀⠀⣠⠰⠾⠷⠈⣦⡤⣿⡀⠀⣠⠉⠐⠀⠾⠶⢾⣇⠀⠀⠷⠀⠀⠀⠀⠻⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡀⠈⠢⠘⣃⣤⠄⠀⠀⠀⢸⣦⡼⢿⡈⠋⣤⡁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⢠⣴⠋⣸⡇⠴⠹⡟⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣀⡟⠻⢦⣴⡇⠀⠀⣈⠁⠀⠀⠀⠀⠀⢀⠈⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠈⠀⠃⠀⠙⢷⠀⠀⠈⢱⡾⠃⠀⠀⠂⠀⠀⠸⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢠⡣⣤⠀⠁⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠛⠀⠠⠤⠀⠀⠀⠛⠋⢷⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣾⣷⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠀⠀⠈⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢠⡀⣰⡄⠀⣄⣈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢦⡀⠀⣀⣠⣤⣤⡤⣤⣤⣾⠋⠉⠁⠀⠀⠀⠠⠀⠀⢰⠤⠤⢤⡄⣉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠙⠲⢤⡄⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣠⠿⠖⠒⠶⠚⠛⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣧⣤⣄⣀⣀⢀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣽⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⠁⣠⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠹⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠃⣀⣄⡠⣶⣶⡀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣨⣶⣦⡻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣴⣿⣿⣿⢏⣼⣿⣿⡿⢱⡿⣱⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⣠⣾⣿⣿⠟⣵⣿⣿⣿⢋⣶⣿⣦⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢏⣴⣿⣿⡿⣫⣾⣿⣿⢟⣵⣿⣿⡿⢋⣾⡟⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⣵⣿⣿⣿⢋⣾⣿⣿⡿⣣⣾⣿⣿⢟⣵⣶⣍⢼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⣿⡿⢋⣾⣿⣿⡿⣡⣿⣿⣿⢏⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⡿⢫⣾⣿⣿⡟⣱⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣴⣿⣿⡿⣫⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣾⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⠿⣟⣩⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠋⠁⠀⠉⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡟⠉⠉⠙⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣀⠀⠀⣠⣾⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⡀⠀⣠⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠛⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠟⠛⠻⠻⢿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠛⠛⠛⠛⢿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣾⣿⣿⠇⠀⠀⢠⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢀⣿⣿⣿⣿⡀⠀⠀⠻⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠲⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣦⣀⣀⣠⣾⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣷⣄⣠⣾⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢠⣤⠀⠰⡶⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠙⠋⢟⣡⢤⣄⡚⠉⠁⠀⠀⣠⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢋⣀⣍⠀⠘⠟⠀⠙⠋⠀⠀⠀⢀⣿⠤⠀⠀⡾⠄⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⣾⠍⠉⠀⠀⡴⣦⢰⣇⣶⣠⢄⢀⡇⣰⡷⢂⡏⣰⡇⡤⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⠸⠇⠀⣯⢾⠵⠧⠿⠤⢺⡇⢿⡜⠷⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢎⡟⠙⠒⠶⠖⠒⠛⠲⠤⠀⠘⠃⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣾⣷⡷⢬⡉⠀⠀⠀⠀⠀⢠⠀⠀⣰⢣⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⡿⢸⢁⣤⠀⡀⠀⡀⢀⡀⣿⢠⠀⣿⢀⡄⣀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣾⡜⣯⣼⣱⣆⣿⢻⣧⣿⡏⢠⡗⣼⣧⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠁⠀⠀⠉⠃⠉⠘⡇⢈⠁⠀⠘⠊⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠂⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⣶⡄⠟⠃
⠀⠀⠀⠀⠀⠀⡴⢂⡄⠀⢸⠏⣤⣿⠀⣰⣦⢀⡀⠀⣀⡀⢠⣶⠄⣠⣼⠃⠀⠀⣠⣼⢿⡆⠁⢶⡆
⠀⠀⠀⠀⠀⠈⠣⠉⢀⡠⢿⢢⠿⡏⠰⣟⣡⠀⣿⡄⢩⣧⢿⣋⣼⡷⣿⡄⠀⣿⡟⠛⢆⡀⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠁⠀⠀⠀⠉⠀⠉⠁⠀⠀⠈⠀⠉⠀⠀⠈⢡⣄⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⣀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠤⣀⣀⣤⣀⣀
⠀⠀⣠⡞⠃⠀⠈⠉⠀⠀⠀⠘⠛⠿⠿⠿⢶⣶⣾⣷⣶⠲⠶⠬⣍⣙⠛⠉⢐⣶⣶⣦⡄
⠀⠀⠻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⣔⢻⢿⣵⣠⣷⣼⣿⣿⢟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣃
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⢿⡿⡿⠿⠯⣿⣿⢿⡷⣯⣭⣭⣥⣴⣾⢟⡄⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⠹⣍⡀⢲⣲⠇⠸⡀⣸⣿⢻⣿⡭⢝⡳⠷⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣀⠀⠀⠀⠀⢻⣆⠀⠂⠹⣿⡶⢁⠀⡼⣰⠾⠛⠚⠯⣿⡷⣆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⠀⠉⠲⢤⣀⡈⠻⠦⠮⣂⣥⠖⣡⠞⣵⠏⠀⠈⠛⠒⣬⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠘⠳⢬⣽⠿⠿⢛⠋⠁⢘⣡⠞⠁⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⣶⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠈⢿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣤⠶⠛⠊⠂⠀⠹⣿⣂⠉⢀⣽⣧⣀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢿⡿⢋⣴⣀⣤⠶⠿⣚⣛⣒⣡⣴⣿⣿⣿⣿⣿⣄⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡟⢱⠋⠀⠈⢛⣩⣴⣶⣿⣿⣿⢟⣿⣿⣿⣿⣷⣾⣿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠹⢶⣶⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⠿⢿⣿⠻⢿⣷⣯⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⣿⣷⣶⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣆⠀⠉⢻⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⡤⠀⠀⠁
⠀⠀⠀⠀⣄⠀⢈⣼⡟⠋⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠻⠿⠿⣿⣿⣿⣿⣦⣄⣴⢦⡀
⠀⠀⠀⠀⢻⣿⡿⠋⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⠛⠉⠁⠈⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡿⠦⡁
⠀⣤⣤⣤⣼⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⣉⠀⠀⠈⠑⢤⡀
⠀⠸⣿⣿⣿⠿⠟⠛⠋⠉⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄
⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⢘⣛⣛⢛⠟⠙⠛⠿⠿⠿⠿⠿⠿⠿⠛⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡎⠀⠀⠉⠻⣿⣿⡦⠀⠀⠀⠙⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢠⡤⠀⠰⣞⢲⡄⣷⡇⣠⠖⡶⠀⠀⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣾⠷⢤⣿⣇⣀⣀⠈⠙⠛⠈⠙⠓⠋⠀⣀⡤⠞⢠⣿⠃⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠈⡀⢠⣿⠀⣼⠇⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⣀⠀⣼⠇⣀⠘⢬
⠀⠀⠀⠀⠀⠀⡴⠚⡁⢸⡏⠀⣿⠔⣹⡆⠀⣠⡶⠋⣷⡟⠀⣴⣦⢺⡿⢀⣿⠊⣨⡇
⠀⠀⠀⠀⠀⠀⡷⠊⠁⠸⣧⠀⠛⠀⣿⠁⢰⣏⠴⠊⣿⢀⠴⣿⠃⣿⢁⠝⠻⣿⠉⠀⢠⠃
⠀⠀⠀⠀⠀⣤⣀⢤⡰⣄⠙⠦⠀⠀⠻⠒⠁⠀⠀⠀⠈⠁⠀⠉⠀⠈⠁⠀⠀⠈⠓⠒⠁⠲⣄
⠀⠀⠀⠀⠀⠀⢈⠗⡿⢺⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⢿⡦
⠀⠀⠀⠀⠀⠘⠁⠈⣁⡋⠀⠀⠀⢰⣿⣆⠀⢀⣠⣤⣤⠀⠀⠀⣠⣤⡄⠀⠀⠀⢀⣤⣄⢠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⣠⣿⣿⣿⢠⣿⣿⠏⠀⢿⣏⠉⣿⣿⡇⠀⠀⠀⢸⣿⣿⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣤⠴⠋⢸⣿⡿⣿⣿⣿⠀⠀⢸⣿⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⠘⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⣼⣿⠇⢹⣿⣿⡀⠀⢸⣿⠀⢹⣿⣷⠀⠀⣼⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⢀⣽⣿⠿⠄⠀⠙⠿⠷⠶⠟⠁⠀⠀⠙⠻⠷⠞⠁⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⡿⠿⠿⠿⠟⢳⣾⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠟⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠈⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⠉⠛⠻⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣄⣀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠤⠤⠀⠀⠘⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⢿⣿⡿⢿⣿⣿⣿⡿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣦⣼⣦⣾⣤⣶⣧⣤⣿⣧⣾⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠋⠙⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠤⠀⢄
⠀⠀⠀⢀⡴⠋⠀⢀⠔⡈⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⣰⠏⠀⠀⡔⠁⡰⠁⠀⠀⠀⠀⠀⡄⢀⡏⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠈⠀⠀⠈⠀⠀⠙⠒⠒⠒⢲⢒⡾⠛⠚⠁⠘⠒⠒⡒⠒⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣸⠛⠙⠓⢦⡀⠀⣀⣭⡐⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⠀⠀⠀⠀⢙⠞⠁⠀⠈⠂⠀⠀⠀⠀⠙⠓⠦⢤⡀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢇⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠞⠀⠀⠀⠀⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠣⢄⣀⡰⠞⢷⠆⠀⠔⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡀⠀⡟⣧⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢴⠒⠒⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⡜⠧⢖⡃⠄⠠⠀⠀⠀⣀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠐⠈⠀⠐⠀⠀⠀⢀⢀⠐⠒⢭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⠀⠠⢠⠀⠀⠂⡀⠀⠑⠚⠁⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠤⠤⠤⢤⢴⡒⢲⢏⠟⠁⠀⠉⢰⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⢳⠌⠃⠃⠠⡄⠀⠀⢸⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣍⡩⠊⠀⠈⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣅⠈⠉⠉⠀⠀⠈⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣄⡜⢡⠜⠻⣧⣠⣆⠀⠀⢀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⣾⣶⣤⣬⣆⠀⠀⠀⠉⢻⠛⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡄⣤⣶⣉⢈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠻⢷⣴⠭⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣄⠀⣠⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡅⣿⣿⣿⠀⠀⠀⢀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇⠻⣿⠟⠀⠀⠀⢸⡇⠹⣧⡀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣤⠀⠀⠀⠀⢸⡇⠀⠘⢿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠈⢻⣧⠹⣟⣋⣭⣭⣉⣛⠻⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣿⡇⢿⣿⣿⣿⣿⣿⣿⣶⣌⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣋⣤⠄⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⢸⣛⣋⣙⣛⠻⣿⣿⣿⣷⡌⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣤⣾⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⢠⣿⡇⣾⣿⣿⣿⣿⣿⣮⢻⣿⣿⣿⡌⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣾⣿⢸⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⣡⣶⣶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⢻⣿⠸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡿⣷⣝⠻⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⣇⢻⣿⣿⣿⣿⡿⢋⣼⣿⣿⣿⢣⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡿⢸⣧⣭⣭⣵⣾⣿⣿⣿⡿⣣⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⢀⣿⠇⣾⣿⣿⣿⣿⣿⡿⠟⣫⣴⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢸⡇⠀⠀⢀⣾⠟⣼⣿⣭⣭⣭⣭⣵⣶⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢸⡇⠀⣠⣿⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⢸⠇⣴⠟⠁⠀⠀⠈⠉⠉⠙⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⡀⢀⡤⣄⠜⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⠈⠋⠉⠋⠀⣀⡠⠤⠒⠚⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⠀⠐⡖⠉⠁⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠀⣵⠀⠀⠀⠀⠀⣠⠞⠐⠖⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢀⠀⠀⠤⠶⢶⠾⢃⡴⠚⠉⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠀⡴⠒⠛⠢⠌⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠲⢤⣤⡁⠃⠀⢸⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛⠯⠓⠀⠙⠛⢷⣤⣀⠳⠄⠀⢲⠖⢤⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⣤⡀⠀⠀⠀⠈⠙⠛⠢⡄⠀⠀⠀⢠⣶⣶⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠹⣷⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣄⢼⠃⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠙⠻⣄⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⠑⡄⢠⡤⠄⠒⠉⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠁⠀⡀⠀⠀⠁⠠⠀⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠰⣄⡀⠀⠈⣀⣴⣬⣧⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠸⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠻⠀⡇⠘⣄⠀⠀⠀⠀⠀⠀⠀⣡⣀⠀⠄⠒⠒⢀⠈⠁⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⢠⣿⠃⠀⠀⠀⠀⣀⡴⠛⠶⠦⠤⠤⠴⡇⠀⠈⢦⠀⠀⠀⠀⢀⡴⢃⠁⣤⠴⢦⡶⠿⣆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⢢⣧⠋⠓⠦⢀⣀⣀⣠⡋⠀⠀⠀⠀⠀⠀⠀⢳⣶⡿⡟⠁⢀⡠⠚⣡⠖⠁⠈⠙⣟⠉⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢓⣖⢿⣿⠀⣄⣀⠎⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢄⠘⠒⠁⠠⠊⠁⠀⠀⠀⠀⠈⠓⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠕⢫⠇⠀⠈⠁⠱⠦⣄⠀⠀⠢⢄⡀⠀⠀⠀⠀⢻⠀⠀⠱⠖⠁⠀⠀⠀⢀⣀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⠉⠁⠈⠉⠉⠉⠒⠢⢤⠀⠙⠫⠉⢉⡭⠟⠣⠉⠉⠁⢈⣉⣭⣭⣥⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⣄⠹⢶⡄⠀⠀⠀⢤⠧⣿⠁⠒⠒⠒⠲⠦⠤⠀⠠⠤⠶⠶⠂⠒⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠻⠤⠤⢴⣇⠀⣀⠀⢸⡖⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠐⠀⠾⠭⠽⠯⠭⠳⠞⠀⠘⠶⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠴⠁⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠠⠤⠤⠤⠄⠀⠀⢀⠂⢄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠄⢀⠀⠶⠬⠭⠥⠴⢛⡀⠦⠀⠀⠀⠈⠐⠈⠀⠭⠤⠴⠶⠆⠀⠉⠀⢂⣀⣁⣀⣀⣀⡱⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⣤⣤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠀⠰⠔⠴⠀⠂⣀⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠇⠀⢐⠊⠁⠀⠀⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁⠀⠀⡸⠀⠀⠀⢠⠞⡈⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇⠀⠉⠘⠀⠊⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢐⠢⢀⣰⠀⠀⠀⠀⡀⠂⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣈⠁⠀⠉⠓⠦⣕⡒⠟⠙⢇⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢨⠋⠉⠉⠉⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠹⣆⣀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠇⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠂⢀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⢀⣠⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠄⠀⠑⠂⢀⠀⠀⠈⠓⠀⠒⠁⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⠢⣀⡀⠑⡀⣜⣆⠀⠀⠀⠀⠀⢁⠔⠒⢺⠱⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⠀⠀⠀⣿⠁⠀⠛⢸⠙⢄⠀⠀⠀⠀⠈⣧⠤⠚⠒⣧⢋⣁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡰⡲⣀⢿⠁⠀⠀⢠⠖⠉⠉⠁⠀⢹⣀⣬⠆⠀⢀⠄⡪⠄⠴⡬⠟⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣮⠀⣧⣄⢀⠟⠗⠈⠀⠀⠀⠀⠀⠸⠛⠍⠢⠊⠡⠊⠀⠀⠀⠑⠤⠖⠁
⠀⠀⠀⠀⣀⡀⢀⡘⢐⡀⢀⣀⠈⣗⠀⢀⡀⠀⠀⣀⠀⠰⢄⠀⠲⠁⠀⠤⠤⠤⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣄⠀⠒⠁⠀⢠⠆⢐⣒⣀⣀⠀⠀⢀⣀⠀⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠸⠤⠤⣇⢀⡀⣶⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠃⠚⠛⠙⠋⠀⠑⠒⠒⠀⠀⠠⠀⠤⠴⠒⠒⠁⠀⠈⠓⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⡀⠠⠀⣀⡀⠀⠀⣠⠖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⡀⠤⢀
⠀⠀⠀⢀⡊⠀⠀⠀⠈⠙⠀⠀⣿⠀⡄⣤⡀⢠⣠⣼⡆⡀⣠⠀⠀⠀⡏⡇⠀⠀⠀⠙⠉⠀⠀⠀⢀⡀
⠀⠀⠠⠉⠌⠀⢀⡾⠛⠳⡄⠀⣿⢠⡇⠉⣇⣏⣿⣼⣸⡏⣟⣸⢷⣆⡏⣄⠀⠀⣠⡴⠖⠒⢶⠀⠡⠋
⠀⢰⡆⠀⠀⠀⠸⡇⠀⣠⠇⠀⠈⣹⠇⠀⠁⠀⠀⠈⠉⠉⣿⠈⠋⠛⠓⠃⣰⣿⣥⡀⠀⠀⡿⠀⢰⡏⢰⡆
⠀⠈⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣀⡽⠴⠶⠶⠶⠤⣤⣀⣀⣉⣀⣀⣀⣀⣤⣿⠛⢻⡇⠀⠀⢣⣠⠞⠀⠀⠁
⠀⠀⠀⣠⠴⠶⠲⠶⠿⣶⣴⡞⠁⣄⣼⠇⠀⢀⠀⠀⠀⠉⣭⡉⠉⠉⠀⢰⡏⠀⢸⡇⠀⠀⢀⣻⡟⠻⢶⣄
⠀⢀⣾⠁⠀⢠⣤⣤⠤⠶⢿⠁⠐⡏⣿⣾⡷⣯⡇⣿⣶⣸⣟⢠⡇⣤⣠⢸⡀⠀⢸⡷⠛⠛⠉⠉⢡⡆⠀⢹⡆
⠀⠸⣿⠀⠀⣈⠳⡀⠀⠀⢸⠀⠀⠷⢘⣀⣁⣀⣉⣉⠘⠛⠗⢈⡇⠛⠛⢈⣡⡴⠟⠁⠀⠀⠀⠀⢸⡁⢀⣼⠃
⠀⠀⠀⣶⣾⣿⡇⢻⠀⠀⢸⣶⣶⣿⠋⠉⠉⠉⠉⠉⠛⠛⠒⠚⠓⠚⠛⠉⠁⠀⠀⣰⢻⡆⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠻⢿⡿⠀⡜⠀⠀⠘⣋⡽⠋⣰⢲⡄⢀⡞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠻⠖⠚⠋⠁⠀⠀⣿⢸⠇⢸⡇⣾⢀⠀⣠⣄⢀⣤⡄⣰⢦⢀⡶⣿⡟⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣫⣄⢸⡿⢣⣟⡇⣿⡉⢸⣧⠀⣿⡞⢸⡇⣿⡇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⣿⣼⣷⣸⣿⣡⢏⣿⡞⣼⡿⠙⠷⠛⠛⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣿⠁⠉⠁⠈⠁⠈⠋⠀⠉⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⢠⠤⢠⣶⣶⠀⡸⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢄⣀⣀⡔⠈⠛⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠂⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⣠⡄⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣁⠀⠠⠀⠀⠀⠙⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣋⢸⠀⠀⣠⣶⢈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠐⢹⣇⠀⠀⠀⠉⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣤⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣘⡿⢃⣰⣘⠷⠀⠺⣦⣀⣙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠰⠀⠀⣀⣀⡀⠀⠴⠤⠴⠚⠛⢻⡟⠉
⠀⣀⠄⠛⠋⠈⠛⣦⡜⣫⠻⠿⣌⠉⣤⣀⣤⣌⣙⣁⠠⠤⣴⡆⡀⠀⠀⠀⢀⣀⣀⡀⠲⠶⠶⠴⠎⡙⠛⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡜⠁⠐⠛⠛⠛⠛⠛⠻⠿⠶⠈⠻⠛⢿⣬⣭⣭⡙⣯⣷⣶⡶⠈⠙⠛⡃⠉⣛⠋⢉⠁⣠⡐⣆⠀⠈⣿⠀⠠⠆
⢿⡜⠲⠄⠀⠀⠀⠤⠀⣀⢠⣄⢀⡀⣾⣏⡉⠴⢾⠿⣿⡿⣿⡆⠾⢆⣡⡄⢩⣦⣱⣾⣿⣧⠼⠷⠀⡾⢶⢶
⠀⠙⠓⠒⠒⠒⠒⠐⠾⠉⠁⠉⠉⠉⠀⠀⠠⢀⠙⢠⡜⣿⣾⠟⠿⠛⠓⠚⠛⠛⠋⠉⠉⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⠀⠀⠀⠀⠀⠀⠀⢡⡄⢃⣻⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢀⠀⠀⠹⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠉⠀⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⡻⢶⣤⣀⠀⠈⠃⠉⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡘
⠀⠀⠀⠀⠀⠀⠀⠰⣛⢁⡞⠁⠀⢀⡝⠛⠢⠤⣀⣸⡿⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠷⣤⣾⣇⠀⠀⠀⠈⢫⠀⠈⢢⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⢠⣾⢻⣿⡝⠁⠀⣷⠟⠻⢻⡶⠀⣤⣼⣄⣴⡟⢿⡽⣧⡀⠀⠀⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⢀⡿⢈⣾⠂⠀⠀⠀⠸⠀⠀⠈⠃⠀⠉⣾⠉⢩⢻⡄⢿⡜⢿⡀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⢸⠀⡿⠀⠀⢰⠀⠀⠀⠀⡄⠀⠀⢹⠄⠇⢻⡀⠘⠀⠘⣤⡎⠻⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠘⢤⡀⠀⣀⣸⡀⢠⠀⢠⡇⠀⣀⢸⡆⠀⢸⠃⠀⢀⣨⣿⡷⠁⠘⣄⡾⠁
⠀⠀⠀⠀⠀⠀⠙⠟⠛⠛⠙⠻⠟⠛⢣⠶⠿⣦⣤⣶⣾⣷⣶⣿⠋⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣀⢂⠀⣀⠀⠰⠷⢀⣾⣿⣿⣹⣿⣯⣽⣿⡄⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢺⠋⠛⠛⠛⠛⠳⣶⢾⣿⣿⡿⠟⠉⠁⢀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣀⣀⠀⣻⣆⡖⢩⣋⠀⠀⠀⣀⣠⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠉⠉⠱⢿⡟⠀⠀⠉⠓⣶⣿⣟⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢈⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣾⡇⠀⠀⠀⠀⢸⣿⣯⣙⣿⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⡇⠀⠀⠀⠀⢸⣿⣝⢮⣽⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⡸⣾⠇⠀⠀⠀⠀⠀⣿⣿⣿⣯⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣦⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⢆⣘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⡿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⢀⣉⣹⡇⠀⠀⠀⠀⠀⠀⠈⣿⣉⡀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠃⢀⣴⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣷⡀⠙⠦

⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⡿⠋⠀⢢⣿⠟⠁
⣿⡿⣿⢋⡅⣡⣼⠟⠁
⣿⣷⣄⢠⣶⠟⠁
⠛⣾⡻⠛⠁
⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢠⡾⠛⠛⢷⡄⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⣰⡟⠀⠀⠀⠀⢻⣆⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣴⠟⠀⢀⣠⣄⡀⠀⠻⣦⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⣼⠏⠀⠀⣾⠋⠙⣷⠀⠀⠹⣧⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⣾⠃⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠘⣷⡀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢀⡾⠃⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠘⢷⡀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⢠⡿⠁⠀⠀⠀⠀⠀⢻⡆⢰⡟⠀⠀⠀⠀⠀⠈⢿⡄⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣠⡟⠁⠀⠀⠀⠀⠀⠀⠈⣻⣟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣆⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠹⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠏⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠋⠁⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⠀⢰⡏⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠲⢄⣸⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⣭⠓⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣄⠀⠀⠀⠀⢿⣗⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⢸⣟⣠⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⡀⠀⠀⠈⢿⣴⣾⣿⣿⣿⣷⠎⡀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡿⢿⣿⣴⣾⢛⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠠⠊⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡶⣿⣿⣿⠟⠟⢀⠀⠀⠀⢀⡿⡀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣷⣿⣿⣿⣶⣾⠋⠀⠀⢀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⣼⡷⠒⠋⠀⠀⠐⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⣎⠉⠉⠉⠙⣛⣿⣿⣿⡟⠛⠃⠶⠀⡀⠁⠈⠀⠀⠀⠀⠀⢀⣀⡀⠤⣿⣧⠀⠀⠀⢀⣀⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠘⠛⠀⢀⡀⠀⣀⡐⢿⡿⡷⣿⣁⣴⣾⣧⣾⣿⣼⡟⠂⠒⠚⢋⡉⠀⠀⠀⠀⢈⡏⠀⠀⠀⠀⠁⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡴⠂⣴⡆⠀⠹⣷⣦⡶⣽⢇⡼⠟⠻⠿⣿⣿⣿⠃⠀⠀⠀⠈⠀⠀⠈⠂⠀⢸⠃⠈⠉⠀⡀⠁⢄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣴⣾⣿⣧⣿⡿⢻⣄⣲⣼⢋⡾⢀⠜⠃⠀⠀⠀⡀⠉⠁⠀⠀⠀⣀⠈⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⡀⠀⣄⠘⢿⣿⣿⣿⣷⣿⣿⣿⣣⣾⡷⠋⣀⢀⠰⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⣨⢟⠀⠈⠀⠂⠀⡀⠠⠊⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠻⢰⣿⣿⣿⠈⣿⣷⣿⣿⣿⠿⠿⣿⡿⣿⣿⣿⠃⠀⠀⠨⠀⠀⠠⠐⠀⡀⠀⠀⠀⠄⠀⠀⣠⠞⠁⢁⡀⠀⠀⠁⣀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⢿⣿⣆⣿⣟⣿⣿⣿⣧⣶⣿⣤⣄⣿⠋⠀⡇⢀⣠⣀⣷⡞⠱⠄⠃⠀⠀⠀⢀⡤⠚⠁⠄⠀⠀⠘⠀⠀⡰⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠶⠴⠚⠃⠀⠘⠌⠣⣷⣾⣿⡟⠿⣛⣿⠉⠍⢿⣿⣿⠳⣦⣾⣿⣿⡟⢿⣛⣦⡔⣿⠀⠠⠜⠉⣠⠤⠀⠀⡀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⢰⡿⣿⣿⣿⡏⠁⠀⠈⠁⠀⠀⠈⣿⠿⡇⢹⡿⣿⠿⡷⠐⣿⣿⠗⠓⠳⢧⠖⣡⣤⡶⠶⠲⣷⢠⡔⣶⠀⡆⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⣿⠁⢀⠀⣀⠁⠀⠈⢿⠁⠀⡇⠀⠀⠀⠀⠀⢹⠀⠀⡄⠃⢻⠃⣇⣈⣿⠃⢠⡤⠦⠠⡠⡤⠌⠠⠠⠈⠿⠸⠼⠷⠰⠇⠀⠀⣾⣵⣀
⠀⠀⠀⠀⠀⢠⠁⠠⠀⠂⠀⠱⠄⠐⢎⠀⠀⠃⠀⠀⣀⣤⠄⠠⠀⣀⢿⡀⠘⠀⢋⡏⣠⠷⠴⣦⢰⡄⠶⢴⡆⠠⠀⠀⠶⠴⠀⠆⠀⠀⠀⡼⠛⠙⣩⠴⠂
⠀⠀⠀⠀⠀⡌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠊⠀⠈⠀⠀⠀⠈⠻⣧⠠⠀⣌⡼⠁⠀⠀⣿⢸⡇⠀⠀⢧⠀⠀⠀⢰⡇⠀⠀⠀⠀⡰⠀⢀⠞⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠃⠀⡰⠀⠀⠀⠀⠀⠀⠀⣿⡄⠰⡟⠀⠀⠀⢸⣿⠀⡇⠀⠀⠈⠀⠀⣰⢿⠃⠀⠀⠀⢰⠃⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠩⢰⠶⠂⢠⠀⡸⢁⡀⠀⠀⠀⡀⠀⠀⠸⠄⠺⠀⠀⠀⠀⢸⣿⠀⠃⠀⠀⠀⢀⠄⡹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠹⢇⠀⠹⣆⠀⠀⢳⡀⠀⠀⣿⡆⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⡴⠋⢀⠃⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠱⣄⠙⢆⠀⠀⠳⡀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸